1 2comment 3 4 Testing some integrals of inverse trigonometric and hyperbolic functions 5 6end; 7 8 9int(asin(x),x); 10 11 12 2 13asin(x)*x + sqrt( - x + 1) 14 15 16int(acos(x),x); 17 18 19 2 20acos(x)*x - sqrt( - x + 1) 21 22 23int(asinh(x),x); 24 25 26 2 27asinh(x)*x - sqrt(x + 1) 28 29 30int(acosh(x),x); 31 32 33 2 34acosh(x)*x - sqrt(x - 1) 35 36 37int(acos(x-1),x); 38 39 40 2 41acos(x - 1)*x - acos(x - 1) - sqrt( - x + 2*x) 42 43 44int(acos(x)*x,x); 45 46 47 2 2 48 2*acos(x)*x + asin(x) - sqrt( - x + 1)*x 49-------------------------------------------- 50 4 51 52 53int(x^2*asin(3*x),x); 54 55 56 3 2 2 2 57 81*asin(3*x)*x + 9*sqrt( - 9*x + 1)*x + 2*sqrt( - 9*x + 1) 58---------------------------------------------------------------- 59 243 60 61 62int(x^2*acos(3*x),x); 63 64 65 3 2 2 2 66 81*acos(3*x)*x - 9*sqrt( - 9*x + 1)*x - 2*sqrt( - 9*x + 1) 67---------------------------------------------------------------- 68 243 69 70 71int(x^2*asinh(3*x),x); 72 73 74 3 2 2 2 75 81*asinh(3*x)*x - 9*sqrt(9*x + 1)*x + 2*sqrt(9*x + 1) 76----------------------------------------------------------- 77 243 78 79 80int(x^2*acosh(3*x),x); 81 82 83 3 2 2 2 84 81*acosh(3*x)*x - 9*sqrt(9*x - 1)*x - 2*sqrt(9*x - 1) 85----------------------------------------------------------- 86 243 87 88 89int(asin(3*x)/x^3,x); 90 91 92 2 93 - asin(3*x) - 3*sqrt( - 9*x + 1)*x 94-------------------------------------- 95 2 96 2*x 97 98 99int(acos(3*x)/x^3,x); 100 101 102 2 103 - acos(3*x) + 3*sqrt( - 9*x + 1)*x 104-------------------------------------- 105 2 106 2*x 107 108 109int(asinh(3*x)/x^3,x); 110 111 112 2 2 113 - asinh(3*x) - 3*sqrt(9*x + 1)*x - 9*x 114------------------------------------------- 115 2 116 2*x 117 118 119int(acosh(3*x)/x^3,x); 120 121 122 2 2 123 - acosh(3*x) - 3*sqrt(9*x - 1)*x - 9*x 124------------------------------------------- 125 2 126 2*x 127 128 129 130% The following is correct, but not optimal 131 132int(x*acos(x-1),x); 133 134 135 2 136(2*acos(x - 1)*x - sqrt(x)*sqrt( - x + 2)*x - 3*sqrt(x)*sqrt( - x + 2) 137 138 sqrt( - x + 2) + sqrt(x)*i 139 - 6*log(----------------------------)*i)/4 140 sqrt(2) 141 142 143df(ws,x); 144 145 146(x*(2*sqrt(x)*sqrt( - x + 2)*acos(x - 1)*x 147 148 - 4*sqrt(x)*sqrt( - x + 2)*acos(x - 1) 149 150 2 2 151 + sqrt(x)*sqrt( - x + 2)*sqrt( - x + 2*x) + x - 2*x))/(2*sqrt(x) 152 153 *sqrt( - x + 2)*(x - 2)) 154 155 156int(x*acos(3*x-2),x); 157 158 159 2 2 160(18*acos(3*x - 2)*x + 9*asin(3*x - 2) - 3*sqrt( - 3*x + 4*x - 1)*sqrt(3)*x 161 162 2 163 - 6*sqrt( - 3*x + 4*x - 1)*sqrt(3) + 8)/36 164 165 166df(ws,x); 167 168 169acos(3*x - 2)*x 170 171 172end; 173 174Tested on x86_64-pc-windows CSL 175Time (counter 1): 0 ms plus GC time: 16 ms 176 177End of Lisp run after 0.00+0.06 seconds 178real 0.20 179user 0.06 180sys 0.03 181