Integrand size = 26, antiderivative size = 465 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {100976 b^2 d^3 x}{4002075 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {256 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \] Output:
100976/4002075*b^2*d^3*x/c^4-50488/12006225*b^2*d^3*x^3/c^2+12622/6670125* b^2*d^3*x^5+9410/1120581*b^2*c^2*d^3*x^7+182/29403*b^2*c^4*d^3*x^9+2/1331* b^2*c^6*d^3*x^11-256/17325*b*d^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^5+ 128/17325*b*d^3*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^3-32/5775*b*d^3 *x^4*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c-16/693*b*d^3*(c^2*x^2+1)^(3/2) *(a+b*arcsinh(c*x))/c^5+4/1155*b*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/ c^5-2/1617*b*d^3*(c^2*x^2+1)^(7/2)*(a+b*arcsinh(c*x))/c^5+8/297*b*d^3*(c^2 *x^2+1)^(9/2)*(a+b*arcsinh(c*x))/c^5-2/121*b*d^3*(c^2*x^2+1)^(11/2)*(a+b*a rcsinh(c*x))/c^5+16/1155*d^3*x^5*(a+b*arcsinh(c*x))^2+8/231*d^3*x^5*(c^2*x ^2+1)*(a+b*arcsinh(c*x))^2+2/33*d^3*x^5*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2 +1/11*d^3*x^5*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2
Time = 0.27 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.64 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (12006225 a^2 c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )-6930 a b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )+2 b^2 c x \left (174940920-29156820 c^2 x^2+13120569 c^4 x^4+58224375 c^6 x^6+42917875 c^8 x^8+10418625 c^{10} x^{10}\right )-6930 b \left (-3465 a c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )+b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )\right ) \text {arcsinh}(c x)+12006225 b^2 c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{13867189875 c^5} \] Input:
Integrate[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*(12006225*a^2*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - 6930*a*b*Sqrt[1 + c^2*x^2]*(50488 - 25244*c^2*x^2 + 18933*c^4*x^4 + 117 625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10) + 2*b^2*c*x*(174940920 - 2 9156820*c^2*x^2 + 13120569*c^4*x^4 + 58224375*c^6*x^6 + 42917875*c^8*x^8 + 10418625*c^10*x^10) - 6930*b*(-3465*a*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^ 4*x^4 + 105*c^6*x^6) + b*Sqrt[1 + c^2*x^2]*(50488 - 25244*c^2*x^2 + 18933* c^4*x^4 + 117625*c^6*x^6 + 111475*c^8*x^8 + 33075*c^10*x^10))*ArcSinh[c*x] + 12006225*b^2*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6)*Ar cSinh[c*x]^2))/(13867189875*c^5)
Time = 4.67 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.49, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {6223, 27, 6219, 27, 1467, 2009, 6223, 6219, 27, 1467, 2009, 6223, 6191, 6219, 27, 2009, 6227, 15, 6227, 15, 6213, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^4 \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {2}{11} b c d^3 \int x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {6}{11} d \int d^2 x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{11} b c d^3 \int x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {6}{11} d^3 \int x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{11} b c d^3 \left (-b c \int \frac {\left (c^2 x^2+1\right )^3 \left (63 c^4 x^4-28 c^2 x^2+8\right )}{693 c^6}dx+\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{11} b c d^3 \left (-\frac {b \int \left (c^2 x^2+1\right )^3 \left (63 c^4 x^4-28 c^2 x^2+8\right )dx}{693 c^5}+\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {2}{11} b c d^3 \left (-\frac {b \int \left (63 c^{10} x^{10}+161 c^8 x^8+113 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{693 c^5}+\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{11} d^3 \int x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {6}{11} d^3 \left (-\frac {2}{9} b c \int x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{9} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c \left (-b c \int \frac {\left (c^2 x^2+1\right )^2 \left (35 c^4 x^4-20 c^2 x^2+8\right )}{315 c^6}dx+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c \left (-\frac {b \int \left (c^2 x^2+1\right )^2 \left (35 c^4 x^4-20 c^2 x^2+8\right )dx}{315 c^5}+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx-\frac {2}{9} b c \left (-\frac {b \int \left (35 c^8 x^8+50 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{315 c^5}+\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (-\frac {2}{7} b c \int x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{7} \int x^4 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c \int x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c \left (-b c \int \frac {15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8}{105 c^6}dx+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{7} b c \left (-\frac {b \int \left (15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{105 c^5}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 c^2}-\frac {b \int x^4dx}{5 c}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {6}{11} d^3 \left (\frac {4}{9} \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )+\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {6}{11} d^3 \left (\frac {1}{9} x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{9} \left (\frac {1}{7} x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))^2-\frac {2}{5} b c \left (\frac {x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {4 \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {b x^5}{25 c}\right )\right )-\frac {2}{7} b c \left (\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6}-\frac {b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )-\frac {2}{9} b c \left (\frac {\left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}-\frac {2 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6}-\frac {b \left (\frac {35 c^8 x^9}{9}+\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\right )-\frac {2}{11} b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{11 c^6}-\frac {2 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^6}+\frac {\left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6}-\frac {b \left (\frac {63 c^{10} x^{11}}{11}+\frac {161 c^8 x^9}{9}+\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{693 c^5}\right )\) |
Input:
Int[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*x^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/11 - (2*b*c*d^3*(-1/693*( b*(8*x - (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (113*c^6*x^7)/7 + (161*c^8*x^9)/9 + (63*c^10*x^11)/11))/c^5 + ((1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7 *c^6) - (2*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^6) + ((1 + c^2*x ^2)^(11/2)*(a + b*ArcSinh[c*x]))/(11*c^6)))/11 + (6*d^3*((x^5*(1 + c^2*x^2 )^2*(a + b*ArcSinh[c*x])^2)/9 - (2*b*c*(-1/315*(b*(8*x - (4*c^2*x^3)/3 + ( 3*c^4*x^5)/5 + (50*c^6*x^7)/7 + (35*c^8*x^9)/9))/c^5 + ((1 + c^2*x^2)^(5/2 )*(a + b*ArcSinh[c*x]))/(5*c^6) - (2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c* x]))/(7*c^6) + ((1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^6)))/9 + (4 *((x^5*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/7 - (2*b*c*(-1/105*(b*(8*x - (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (15*c^6*x^7)/7))/c^5 + ((1 + c^2*x^2)^(3/2 )*(a + b*ArcSinh[c*x]))/(3*c^6) - (2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c* x]))/(5*c^6) + ((1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^6)))/7 + (2 *((x^5*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*(-1/25*(b*x^5)/c + (x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) - (4*(-1/9*(b*x^3)/c + (x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) - (2*(-((b*x)/c) + (Sqrt[1 + c^2 *x^2]*(a + b*ArcSinh[c*x]))/c^2))/(3*c^2)))/(5*c^2)))/5))/7))/9))/11
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 2.66 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.12
| method | result | size |
| parts | \(d^{3} a^{2} \left (\frac {1}{11} c^{6} x^{11}+\frac {1}{3} c^{4} x^{9}+\frac {3}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{3} b^{2} \left (\frac {x^{3} c^{3} \left (c^{2} x^{2}+1\right )^{4} \operatorname {arcsinh}\left (x c \right )^{2}}{11}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{1155}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{1155}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{1155}+\frac {382986368 x c}{13867189875}-\frac {428 x c \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {148174 x c \left (c^{2} x^{2}+1\right )^{3}}{110937519}-\frac {5487704 x c \left (c^{2} x^{2}+1\right )^{2}}{4622396625}+\frac {34 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{5}}{1331}-\frac {606416 x c \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}\right )}{c^{5}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{11} c^{11}}{11}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {6311 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {4705 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {91 \sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{3267}-\frac {x^{10} c^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(519\) |
| derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{11} x^{11} c^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+d^{3} b^{2} \left (\frac {x^{3} c^{3} \left (c^{2} x^{2}+1\right )^{4} \operatorname {arcsinh}\left (x c \right )^{2}}{11}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{1155}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{1155}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{1155}+\frac {382986368 x c}{13867189875}-\frac {428 x c \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {148174 x c \left (c^{2} x^{2}+1\right )^{3}}{110937519}-\frac {5487704 x c \left (c^{2} x^{2}+1\right )^{2}}{4622396625}+\frac {34 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{5}}{1331}-\frac {606416 x c \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{11} c^{11}}{11}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {6311 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {4705 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {91 \sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{3267}-\frac {x^{10} c^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(520\) |
| default | \(\frac {d^{3} a^{2} \left (\frac {1}{11} x^{11} c^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} x^{7} c^{7}+\frac {1}{5} x^{5} c^{5}\right )+d^{3} b^{2} \left (\frac {x^{3} c^{3} \left (c^{2} x^{2}+1\right )^{4} \operatorname {arcsinh}\left (x c \right )^{2}}{11}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \operatorname {arcsinh}\left (x c \right )^{2} x c}{1155}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{1155}-\frac {4 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {32 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{1155}+\frac {382986368 x c}{13867189875}-\frac {428 x c \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {148174 x c \left (c^{2} x^{2}+1\right )^{3}}{110937519}-\frac {5487704 x c \left (c^{2} x^{2}+1\right )^{2}}{4622396625}+\frac {34 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{5}}{1331}-\frac {606416 x c \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{11} c^{11}}{11}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{9} c^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}-\frac {6311 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {4705 x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {91 \sqrt {c^{2} x^{2}+1}\, x^{8} c^{8}}{3267}-\frac {x^{10} c^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(520\) |
| orering | \(\frac {\left (3448564875 c^{14} x^{14}+16454567500 c^{12} x^{12}+29885660250 c^{10} x^{10}+23335495700 c^{8} x^{8}+3719665587 c^{6} x^{6}+16269505560 c^{4} x^{4}+15161546400 c^{2} x^{2}+4198582080\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{13867189875 x \,c^{6} \left (c^{2} x^{2}+1\right )^{4}}-\frac {\left (312558750 c^{12} x^{12}+1399654375 c^{10} x^{10}+2243437625 c^{8} x^{8}+1188259281 c^{6} x^{6}-470882643 c^{4} x^{4}+3178093380 c^{2} x^{2}+1574468280\right ) \left (4 x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+6 x^{5} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{13867189875 x^{4} c^{6} \left (c^{2} x^{2}+1\right )^{3}}+\frac {\left (10418625 c^{10} x^{10}+42917875 c^{8} x^{8}+58224375 c^{6} x^{6}+13120569 c^{4} x^{4}-29156820 c^{2} x^{2}+174940920\right ) \left (12 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+54 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {16 x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+24 x^{6} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}+\frac {24 x^{5} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{5} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{13867189875 x^{3} c^{6} \left (c^{2} x^{2}+1\right )^{2}}\) | \(571\) |
Input:
int(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
d^3*a^2*(1/11*c^6*x^11+1/3*c^4*x^9+3/7*c^2*x^7+1/5*x^5)+d^3*b^2/c^5*(1/11* x^3*c^3*(c^2*x^2+1)^4*arcsinh(x*c)^2-1/33*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^4 +16/1155*arcsinh(x*c)^2*x*c+1/231*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^3+2/385*a rcsinh(x*c)^2*x*c*(c^2*x^2+1)^2+8/1155*arcsinh(x*c)^2*x*c*(c^2*x^2+1)-4/19 25*arcsinh(x*c)*(c^2*x^2+1)^(5/2)-32/1155*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+3 82986368/13867189875*x*c-428/323433*x*c*(c^2*x^2+1)^4-2/1617*arcsinh(x*c)* (c^2*x^2+1)^(7/2)-148174/110937519*x*c*(c^2*x^2+1)^3-5487704/4622396625*x* c*(c^2*x^2+1)^2+34/3267*arcsinh(x*c)*(c^2*x^2+1)^(9/2)-2/121*arcsinh(x*c)* x^2*c^2*(c^2*x^2+1)^(9/2)+2/1331*x*c*(c^2*x^2+1)^5-606416/13867189875*x*c* (c^2*x^2+1)-16/3465*arcsinh(x*c)*(c^2*x^2+1)^(3/2))+2*d^3*a*b/c^5*(1/11*ar csinh(x*c)*x^11*c^11+1/3*arcsinh(x*c)*x^9*c^9+3/7*arcsinh(x*c)*c^7*x^7+1/5 *arcsinh(x*c)*x^5*c^5-6311/1334025*x^4*c^4*(c^2*x^2+1)^(1/2)+25244/4002075 *x^2*c^2*(c^2*x^2+1)^(1/2)-50488/4002075*(c^2*x^2+1)^(1/2)-4705/160083*x^6 *c^6*(c^2*x^2+1)^(1/2)-91/3267*(c^2*x^2+1)^(1/2)*x^8*c^8-1/121*x^10*c^10*( c^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.95 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {10418625 \, {\left (121 \, a^{2} + 2 \, b^{2}\right )} c^{11} d^{3} x^{11} + 471625 \, {\left (9801 \, a^{2} + 182 \, b^{2}\right )} c^{9} d^{3} x^{9} + 12375 \, {\left (480249 \, a^{2} + 9410 \, b^{2}\right )} c^{7} d^{3} x^{7} + 2079 \, {\left (1334025 \, a^{2} + 12622 \, b^{2}\right )} c^{5} d^{3} x^{5} - 58313640 \, b^{2} c^{3} d^{3} x^{3} + 349881840 \, b^{2} c d^{3} x + 12006225 \, {\left (105 \, b^{2} c^{11} d^{3} x^{11} + 385 \, b^{2} c^{9} d^{3} x^{9} + 495 \, b^{2} c^{7} d^{3} x^{7} + 231 \, b^{2} c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6930 \, {\left (363825 \, a b c^{11} d^{3} x^{11} + 1334025 \, a b c^{9} d^{3} x^{9} + 1715175 \, a b c^{7} d^{3} x^{7} + 800415 \, a b c^{5} d^{3} x^{5} - {\left (33075 \, b^{2} c^{10} d^{3} x^{10} + 111475 \, b^{2} c^{8} d^{3} x^{8} + 117625 \, b^{2} c^{6} d^{3} x^{6} + 18933 \, b^{2} c^{4} d^{3} x^{4} - 25244 \, b^{2} c^{2} d^{3} x^{2} + 50488 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6930 \, {\left (33075 \, a b c^{10} d^{3} x^{10} + 111475 \, a b c^{8} d^{3} x^{8} + 117625 \, a b c^{6} d^{3} x^{6} + 18933 \, a b c^{4} d^{3} x^{4} - 25244 \, a b c^{2} d^{3} x^{2} + 50488 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{13867189875 \, c^{5}} \] Input:
integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/13867189875*(10418625*(121*a^2 + 2*b^2)*c^11*d^3*x^11 + 471625*(9801*a^2 + 182*b^2)*c^9*d^3*x^9 + 12375*(480249*a^2 + 9410*b^2)*c^7*d^3*x^7 + 2079 *(1334025*a^2 + 12622*b^2)*c^5*d^3*x^5 - 58313640*b^2*c^3*d^3*x^3 + 349881 840*b^2*c*d^3*x + 12006225*(105*b^2*c^11*d^3*x^11 + 385*b^2*c^9*d^3*x^9 + 495*b^2*c^7*d^3*x^7 + 231*b^2*c^5*d^3*x^5)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6930*(363825*a*b*c^11*d^3*x^11 + 1334025*a*b*c^9*d^3*x^9 + 1715175*a*b*c ^7*d^3*x^7 + 800415*a*b*c^5*d^3*x^5 - (33075*b^2*c^10*d^3*x^10 + 111475*b^ 2*c^8*d^3*x^8 + 117625*b^2*c^6*d^3*x^6 + 18933*b^2*c^4*d^3*x^4 - 25244*b^2 *c^2*d^3*x^2 + 50488*b^2*d^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6930*(33075*a*b*c^10*d^3*x^10 + 111475*a*b*c^8*d^3*x^8 + 117625*a*b* c^6*d^3*x^6 + 18933*a*b*c^4*d^3*x^4 - 25244*a*b*c^2*d^3*x^2 + 50488*a*b*d^ 3)*sqrt(c^2*x^2 + 1))/c^5
Time = 3.20 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.51 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x**4*(c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**6*d**3*x**11/11 + a**2*c**4*d**3*x**9/3 + 3*a**2*c**2*d **3*x**7/7 + a**2*d**3*x**5/5 + 2*a*b*c**6*d**3*x**11*asinh(c*x)/11 - 2*a* b*c**5*d**3*x**10*sqrt(c**2*x**2 + 1)/121 + 2*a*b*c**4*d**3*x**9*asinh(c*x )/3 - 182*a*b*c**3*d**3*x**8*sqrt(c**2*x**2 + 1)/3267 + 6*a*b*c**2*d**3*x* *7*asinh(c*x)/7 - 9410*a*b*c*d**3*x**6*sqrt(c**2*x**2 + 1)/160083 + 2*a*b* d**3*x**5*asinh(c*x)/5 - 12622*a*b*d**3*x**4*sqrt(c**2*x**2 + 1)/(1334025* c) + 50488*a*b*d**3*x**2*sqrt(c**2*x**2 + 1)/(4002075*c**3) - 100976*a*b*d **3*sqrt(c**2*x**2 + 1)/(4002075*c**5) + b**2*c**6*d**3*x**11*asinh(c*x)** 2/11 + 2*b**2*c**6*d**3*x**11/1331 - 2*b**2*c**5*d**3*x**10*sqrt(c**2*x**2 + 1)*asinh(c*x)/121 + b**2*c**4*d**3*x**9*asinh(c*x)**2/3 + 182*b**2*c**4 *d**3*x**9/29403 - 182*b**2*c**3*d**3*x**8*sqrt(c**2*x**2 + 1)*asinh(c*x)/ 3267 + 3*b**2*c**2*d**3*x**7*asinh(c*x)**2/7 + 9410*b**2*c**2*d**3*x**7/11 20581 - 9410*b**2*c*d**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/160083 + b**2 *d**3*x**5*asinh(c*x)**2/5 + 12622*b**2*d**3*x**5/6670125 - 12622*b**2*d** 3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(1334025*c) - 50488*b**2*d**3*x**3/( 12006225*c**2) + 50488*b**2*d**3*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(4002 075*c**3) + 100976*b**2*d**3*x/(4002075*c**4) - 100976*b**2*d**3*sqrt(c**2 *x**2 + 1)*asinh(c*x)/(4002075*c**5), Ne(c, 0)), (a**2*d**3*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 1109 vs. \(2 (413) = 826\).
Time = 0.07 (sec) , antiderivative size = 1109, normalized size of antiderivative = 2.38 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Too large to display} \] Input:
integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/11*b^2*c^6*d^3*x^11*arcsinh(c*x)^2 + 1/11*a^2*c^6*d^3*x^11 + 1/3*b^2*c^4 *d^3*x^9*arcsinh(c*x)^2 + 1/3*a^2*c^4*d^3*x^9 + 3/7*b^2*c^2*d^3*x^7*arcsin h(c*x)^2 + 3/7*a^2*c^2*d^3*x^7 + 2/7623*(693*x^11*arcsinh(c*x) - (63*sqrt( c^2*x^2 + 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(c^2*x^2 + 1 )*x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*x^4/c^8 + 128*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c)*a*b*c^6*d^3 - 2/26413695*(3465*(63*sqrt(c ^2*x^2 + 1)*x^10/c^2 - 70*sqrt(c^2*x^2 + 1)*x^8/c^4 + 80*sqrt(c^2*x^2 + 1) *x^6/c^6 - 96*sqrt(c^2*x^2 + 1)*x^4/c^8 + 128*sqrt(c^2*x^2 + 1)*x^2/c^10 - 256*sqrt(c^2*x^2 + 1)/c^12)*c*arcsinh(c*x) - (19845*c^10*x^11 - 26950*c^8 *x^9 + 39600*c^6*x^7 - 66528*c^4*x^5 + 147840*c^2*x^3 - 887040*x)/c^10)*b^ 2*c^6*d^3 + 1/5*b^2*d^3*x^5*arcsinh(c*x)^2 + 2/945*(315*x^9*arcsinh(c*x) - (35*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sqrt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^ 2*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/ c^10)*c)*a*b*c^4*d^3 - 2/297675*(315*(35*sqrt(c^2*x^2 + 1)*x^8/c^2 - 40*sq rt(c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(c^2*x^2 + 1)*x^4/c^6 - 64*sqrt(c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(c^2*x^2 + 1)/c^10)*c*arcsinh(c*x) - (1225*c^8*x^9 - 1800*c^6*x^7 + 3024*c^4*x^5 - 6720*c^2*x^3 + 40320*x)/c^8)*b^2*c^4*d^3 + 1/5*a^2*d^3*x^5 + 6/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^ 2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^ 2*x^2 + 1)/c^8)*c)*a*b*c^2*d^3 - 2/8575*(105*(5*sqrt(c^2*x^2 + 1)*x^6/c...
Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int(x^4*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3,x)
Output:
int(x^4*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3, x)
\[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{3} \left (727650 \mathit {asinh} \left (c x \right ) a b \,c^{11} x^{11}+2668050 \mathit {asinh} \left (c x \right ) a b \,c^{9} x^{9}+3430350 \mathit {asinh} \left (c x \right ) a b \,c^{7} x^{7}+1600830 \mathit {asinh} \left (c x \right ) a b \,c^{5} x^{5}-66150 \sqrt {c^{2} x^{2}+1}\, a b \,c^{10} x^{10}-222950 \sqrt {c^{2} x^{2}+1}\, a b \,c^{8} x^{8}-235250 \sqrt {c^{2} x^{2}+1}\, a b \,c^{6} x^{6}-37866 \sqrt {c^{2} x^{2}+1}\, a b \,c^{4} x^{4}+50488 \sqrt {c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-100976 \sqrt {c^{2} x^{2}+1}\, a b +4002075 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{10}d x \right ) b^{2} c^{11}+12006225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{8}d x \right ) b^{2} c^{9}+12006225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+4002075 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+363825 a^{2} c^{11} x^{11}+1334025 a^{2} c^{9} x^{9}+1715175 a^{2} c^{7} x^{7}+800415 a^{2} c^{5} x^{5}\right )}{4002075 c^{5}} \] Input:
int(x^4*(c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2,x)
Output:
(d**3*(727650*asinh(c*x)*a*b*c**11*x**11 + 2668050*asinh(c*x)*a*b*c**9*x** 9 + 3430350*asinh(c*x)*a*b*c**7*x**7 + 1600830*asinh(c*x)*a*b*c**5*x**5 - 66150*sqrt(c**2*x**2 + 1)*a*b*c**10*x**10 - 222950*sqrt(c**2*x**2 + 1)*a*b *c**8*x**8 - 235250*sqrt(c**2*x**2 + 1)*a*b*c**6*x**6 - 37866*sqrt(c**2*x* *2 + 1)*a*b*c**4*x**4 + 50488*sqrt(c**2*x**2 + 1)*a*b*c**2*x**2 - 100976*s qrt(c**2*x**2 + 1)*a*b + 4002075*int(asinh(c*x)**2*x**10,x)*b**2*c**11 + 1 2006225*int(asinh(c*x)**2*x**8,x)*b**2*c**9 + 12006225*int(asinh(c*x)**2*x **6,x)*b**2*c**7 + 4002075*int(asinh(c*x)**2*x**4,x)*b**2*c**5 + 363825*a* *2*c**11*x**11 + 1334025*a**2*c**9*x**9 + 1715175*a**2*c**7*x**7 + 800415* a**2*c**5*x**5))/(4002075*c**5)