Integrand size = 23, antiderivative size = 439 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {2 b^2 (a+b \arcsin (c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \arcsin (c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b (a+b \arcsin (c+d x))^3 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{3 d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {2 i b^2 (a+b \arcsin (c+d x))^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )}{d e^4}-\frac {4 i b^4 \operatorname {PolyLog}\left (4,-e^{i \arcsin (c+d x)}\right )}{d e^4}+\frac {4 i b^4 \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )}{d e^4} \]
-2*b^2*(a+b*arcsin(d*x+c))^2/d/e^4/(d*x+c)-1/3*(a+b*arcsin(d*x+c))^4/d/e^4 /(d*x+c)^3-8*b^3*(a+b*arcsin(d*x+c))*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2) )/d/e^4-4/3*b*(a+b*arcsin(d*x+c))^3*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2)) /d/e^4+4*I*b^4*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4+2*I*b^2*(a+ b*arcsin(d*x+c))^2*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4-4*I*b^4 *polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-2*I*b^2*(a+b*arcsin(d*x+c) )^2*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-4*b^3*(a+b*arcsin(d*x+c ))*polylog(3,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4+4*b^3*(a+b*arcsin(d*x+c ))*polylog(3,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^4-4*I*b^4*polylog(4,-I*(d* x+c)-(1-(d*x+c)^2)^(1/2))/d/e^4+4*I*b^4*polylog(4,I*(d*x+c)+(1-(d*x+c)^2)^ (1/2))/d/e^4-2/3*b*(a+b*arcsin(d*x+c))^3*(1-(d*x+c)^2)^(1/2)/d/e^4/(d*x+c) ^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1274\) vs. \(2(439)=878\).
Time = 11.49 (sec) , antiderivative size = 1274, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx =\text {Too large to display} \]
-1/3*a^4/(d*e^4*(c + d*x)^3) + (a^2*b^2*((8*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*(2 + 4*ArcSin[c + d*x]^2 - 2*Cos[2*ArcSin[c + d*x]] - 3*(c + d*x)*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] + 3*(c + d*x)*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] + (4*I)*(c + d*x)^3*PolyLog[2, E^(I *ArcSin[c + d*x])] + 2*ArcSin[c + d*x]*Sin[2*ArcSin[c + d*x]] + ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]] - ArcSin[c + d *x]*Log[1 + E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]]))/(c + d*x)^3))/ (4*d*e^4) + (a*b^3*(-24*ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[ c + d*x]^3*Cot[ArcSin[c + d*x]/2] - 6*ArcSin[c + d*x]^2*Csc[ArcSin[c + d*x ]/2]^2 - (c + d*x)*ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^4 + 24*ArcSin[ c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] - 24*ArcSin[c + d*x]^2*Log[1 + E ^(I*ArcSin[c + d*x])] + 48*Log[Tan[ArcSin[c + d*x]/2]] + (48*I)*ArcSin[c + d*x]*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (48*I)*ArcSin[c + d*x]*PolyLog[ 2, E^(I*ArcSin[c + d*x])] - 48*PolyLog[3, -E^(I*ArcSin[c + d*x])] + 48*Pol yLog[3, E^(I*ArcSin[c + d*x])] + 6*ArcSin[c + d*x]^2*Sec[ArcSin[c + d*x]/2 ]^2 - (16*ArcSin[c + d*x]^3*Sin[ArcSin[c + d*x]/2]^4)/(c + d*x)^3 - 24*Arc Sin[c + d*x]*Tan[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Tan[ArcSin[c + d *x]/2]))/(12*d*e^4) + (b^4*((-2*I)*Pi^4 + (4*I)*ArcSin[c + d*x]^4 - 24*Arc Sin[c + d*x]^2*Cot[ArcSin[c + d*x]/2] - 2*ArcSin[c + d*x]^4*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^2 - ((c + d*x)*Ar...
Time = 1.62 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {5304, 27, 5138, 5204, 5138, 5218, 3042, 4671, 2715, 2838, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^4}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^4}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {\frac {4}{3} b \int \frac {(a+b \arcsin (c+d x))^3}{(c+d x)^3 \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \int \frac {(a+b \arcsin (c+d x))^2}{(c+d x)^2}d(c+d x)+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^3}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \left (2 b \int \frac {a+b \arcsin (c+d x)}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {(a+b \arcsin (c+d x))^2}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^3}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \left (2 b \int \frac {a+b \arcsin (c+d x)}{c+d x}d\arcsin (c+d x)-\frac {(a+b \arcsin (c+d x))^2}{c+d x}\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^3}{c+d x}d\arcsin (c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \left (2 b \int (a+b \arcsin (c+d x)) \csc (\arcsin (c+d x))d\arcsin (c+d x)-\frac {(a+b \arcsin (c+d x))^2}{c+d x}\right )+\frac {1}{2} \int (a+b \arcsin (c+d x))^3 \csc (\arcsin (c+d x))d\arcsin (c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-b \int \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+b \int \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )+\frac {1}{2} \left (-3 b \int (a+b \arcsin (c+d x))^2 \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+3 b \int (a+b \arcsin (c+d x))^2 \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (-3 b \int (a+b \arcsin (c+d x))^2 \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+3 b \int (a+b \arcsin (c+d x))^2 \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-i b \int e^{-i \arcsin (c+d x)} \log \left (1+e^{i \arcsin (c+d x)}\right )de^{i \arcsin (c+d x)}+i b \int e^{-i \arcsin (c+d x)} \log (-c-d x+1)de^{i \arcsin (c+d x)}-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (-3 b \int (a+b \arcsin (c+d x))^2 \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+3 b \int (a+b \arcsin (c+d x))^2 \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (3 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \int (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-3 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \int (a+b \arcsin (c+d x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (3 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \left (i b \int \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-i \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )-3 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \left (i b \int \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-i \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {1}{2} \left (-3 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \left (b \int e^{-i \arcsin (c+d x)} \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right )de^{i \arcsin (c+d x)}-i \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )+3 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \left (b \int e^{-i \arcsin (c+d x)} \operatorname {PolyLog}(3,-c-d x)de^{i \arcsin (c+d x)}-i \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {3}{2} b \left (-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )+\frac {1}{2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^3-3 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \left (b \operatorname {PolyLog}\left (4,e^{i \arcsin (c+d x)}\right )-i \operatorname {PolyLog}\left (3,e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )+3 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))^2-2 i b \left (b \operatorname {PolyLog}(4,-c-d x)-i \operatorname {PolyLog}\left (3,-e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )\right )\right )-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
(-1/3*(a + b*ArcSin[c + d*x])^4/(c + d*x)^3 + (4*b*(-1/2*(Sqrt[1 - (c + d* x)^2]*(a + b*ArcSin[c + d*x])^3)/(c + d*x)^2 + (3*b*(-((a + b*ArcSin[c + d *x])^2/(c + d*x)) + 2*b*(-2*(a + b*ArcSin[c + d*x])*ArcTanh[E^(I*ArcSin[c + d*x])] - I*b*PolyLog[2, E^(I*ArcSin[c + d*x])] + I*b*PolyLog[2, -c - d*x ])))/2 + (-2*(a + b*ArcSin[c + d*x])^3*ArcTanh[E^(I*ArcSin[c + d*x])] - 3* b*(I*(a + b*ArcSin[c + d*x])^2*PolyLog[2, E^(I*ArcSin[c + d*x])] - (2*I)*b *((-I)*(a + b*ArcSin[c + d*x])*PolyLog[3, E^(I*ArcSin[c + d*x])] + b*PolyL og[4, E^(I*ArcSin[c + d*x])])) + 3*b*(I*(a + b*ArcSin[c + d*x])^2*PolyLog[ 2, -E^(I*ArcSin[c + d*x])] - (2*I)*b*((-I)*(a + b*ArcSin[c + d*x])*PolyLog [3, -E^(I*ArcSin[c + d*x])] + b*PolyLog[4, -c - d*x])))/2))/3)/(d*e^4)
3.3.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Time = 1.38 (sec) , antiderivative size = 1009, normalized size of antiderivative = 2.30
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1009\) |
| default | \(\text {Expression too large to display}\) | \(1009\) |
| parts | \(\text {Expression too large to display}\) | \(1020\) |
1/d*(-1/3*a^4/e^4/(d*x+c)^3+b^4/e^4*(-1/3/(d*x+c)^3*arcsin(d*x+c)^2*(2*arc sin(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+arcsin(d*x+c)^2+6*(d*x+c)^2)-2/3*ar csin(d*x+c)^3*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+2*I*arcsin(d*x+c)^2*poly log(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-4*arcsin(d*x+c)*polylog(3,-I*(d*x+c) -(1-(d*x+c)^2)^(1/2))-4*I*polylog(4,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))+2/3*ar csin(d*x+c)^3*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-2*I*arcsin(d*x+c)^2*poly log(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+4*arcsin(d*x+c)*polylog(3,I*(d*x+c)+( 1-(d*x+c)^2)^(1/2))+4*I*polylog(4,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-4*arcsin( d*x+c)*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+4*I*polylog(2,-I*(d*x+c)-(1-(d* x+c)^2)^(1/2))+4*arcsin(d*x+c)*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-4*I*pol ylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2)))+4*a*b^3/e^4*(-1/6/(d*x+c)^3*arcsin( d*x+c)*(3*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+2*arcsin(d*x+c)^2+6*(d *x+c)^2)+1/2*arcsin(d*x+c)^2*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-I*arcsin( d*x+c)*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+polylog(3,I*(d*x+c)+(1-(d* x+c)^2)^(1/2))-1/2*arcsin(d*x+c)^2*ln(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))+I*a rcsin(d*x+c)*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-polylog(3,-I*(d*x+c )-(1-(d*x+c)^2)^(1/2))-2*arctanh(I*(d*x+c)+(1-(d*x+c)^2)^(1/2)))+6*a^2*b^2 /e^4*(-1/3*(arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2)*(d*x+c)+arcsin(d*x+c)^2+(d*x +c)^2)/(d*x+c)^3+1/3*arcsin(d*x+c)*ln(1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-1/3 *I*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-1/3*arcsin(d*x+c)*ln(1+I*(d...
\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*ar csin(d*x + c)^2 + 4*a^3*b*arcsin(d*x + c) + a^4)/(d^4*e^4*x^4 + 4*c*d^3*e^ 4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)
\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{4}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a^{3} b \operatorname {asin}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(Integral(a**4/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 4*x**4), x) + Integral(b**4*asin(c + d*x)**4/(c**4 + 4*c**3*d*x + 6*c**2*d **2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(4*a*b**3*asin(c + d*x )**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x ) + Integral(6*a**2*b**2*asin(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2 *x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(4*a**3*b*asin(c + d*x)/( c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/e** 4
\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*a^4/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1 /3*(b^4*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + 3*(d^4* e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4)*integrate(2/3*(2* (b^4*d*x + b^4*c)*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*arctan2(d*x + c, sq rt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 - 6*(a*b^3*d^2*x^2 + 2*a*b^3*c*d*x + a*b^3*c^2 - a*b^3)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)) ^3 - 9*(a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + a^2*b^2*c^2 - a^2*b^2)*arctan2 (d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2 - 6*(a^3*b*d^2*x^2 + 2*a ^3*b*c*d*x + a^3*b*c^2 - a^3*b)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d *x - c + 1)))/(d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + (15*c^2 - 1)*d^4*e^4*x^4 + 4*(5*c^3 - c)*d^3*e^4*x^3 + 3*(5*c^4 - 2*c^2)*d^2*e^4*x^2 + 2*(3*c^5 - 2*c ^3)*d*e^4*x + (c^6 - c^4)*e^4), x))/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2 *d^2*e^4*x + c^3*d*e^4)
\[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^4}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]