Integrand size = 23, antiderivative size = 385 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2}{d e^4 (c+d x)}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 d e^4 (c+d x)^3}-\frac {8 b^3 (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {2 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {4 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {4 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \]
-2*b^2*(a+b*arcsinh(d*x+c))^2/d/e^4/(d*x+c)-1/3*(a+b*arcsinh(d*x+c))^4/d/e ^4/(d*x+c)^3-8*b^3*(a+b*arcsinh(d*x+c))*arctanh(d*x+c+(1+(d*x+c)^2)^(1/2)) /d/e^4+4/3*b*(a+b*arcsinh(d*x+c))^3*arctanh(d*x+c+(1+(d*x+c)^2)^(1/2))/d/e ^4-4*b^4*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4+2*b^2*(a+b*arcsinh(d* x+c))^2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4+4*b^4*polylog(2,d*x+c+ (1+(d*x+c)^2)^(1/2))/d/e^4-2*b^2*(a+b*arcsinh(d*x+c))^2*polylog(2,d*x+c+(1 +(d*x+c)^2)^(1/2))/d/e^4-4*b^3*(a+b*arcsinh(d*x+c))*polylog(3,-d*x-c-(1+(d *x+c)^2)^(1/2))/d/e^4+4*b^3*(a+b*arcsinh(d*x+c))*polylog(3,d*x+c+(1+(d*x+c )^2)^(1/2))/d/e^4+4*b^4*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4-4*b^4* polylog(4,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4-2/3*b*(a+b*arcsinh(d*x+c))^3*(1 +(d*x+c)^2)^(1/2)/d/e^4/(d*x+c)^2
Leaf count is larger than twice the leaf count of optimal. \(1198\) vs. \(2(385)=770\).
Time = 8.24 (sec) , antiderivative size = 1198, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx =\text {Too large to display} \]
-1/3*a^4/(d*e^4*(c + d*x)^3) + (a^2*b^2*(-8*PolyLog[2, -E^(-ArcSinh[c + d* x])] - (2*(-2 + 4*ArcSinh[c + d*x]^2 + 2*Cosh[2*ArcSinh[c + d*x]] - 3*(c + d*x)*ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])] + 3*(c + d*x)*ArcSin h[c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2, E^(-A rcSinh[c + d*x])] + 2*ArcSinh[c + d*x]*Sinh[2*ArcSinh[c + d*x]] + ArcSinh[ c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]] - ArcSinh [c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]]))/(c + d *x)^3))/(4*d*e^4) + (a*b^3*(-24*ArcSinh[c + d*x]*Coth[ArcSinh[c + d*x]/2] + 4*ArcSinh[c + d*x]^3*Coth[ArcSinh[c + d*x]/2] - 6*ArcSinh[c + d*x]^2*Csc h[ArcSinh[c + d*x]/2]^2 - (c + d*x)*ArcSinh[c + d*x]^3*Csch[ArcSinh[c + d* x]/2]^4 - 24*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] + 24*ArcSin h[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 48*Log[Tanh[ArcSinh[c + d*x] /2]] - 48*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*x])] + 48*ArcSinh [c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] - 48*PolyLog[3, -E^(-ArcSinh[c + d*x])] + 48*PolyLog[3, E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]^2*Se ch[ArcSinh[c + d*x]/2]^2 - (16*ArcSinh[c + d*x]^3*Sinh[ArcSinh[c + d*x]/2] ^4)/(c + d*x)^3 + 24*ArcSinh[c + d*x]*Tanh[ArcSinh[c + d*x]/2] - 4*ArcSinh [c + d*x]^3*Tanh[ArcSinh[c + d*x]/2]))/(12*d*e^4) + (b^4*(-2*Pi^4 + 4*ArcS inh[c + d*x]^4 - 24*ArcSinh[c + d*x]^2*Coth[ArcSinh[c + d*x]/2] + 2*ArcSin h[c + d*x]^4*Coth[ArcSinh[c + d*x]/2] - 4*ArcSinh[c + d*x]^3*Csch[ArcSi...
Result contains complex when optimal does not.
Time = 1.72 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.86, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6274, 27, 6191, 6224, 6191, 6231, 3042, 26, 4670, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {\frac {4}{3} b \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x)^3 \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x)^2}d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \left (2 b \int \frac {a+b \text {arcsinh}(c+d x)}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {\frac {4}{3} b \left (\frac {3}{2} b \left (2 b \int \frac {a+b \text {arcsinh}(c+d x)}{c+d x}d\text {arcsinh}(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}d\text {arcsinh}(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 b \int i (a+b \text {arcsinh}(c+d x)) \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)\right )-\frac {1}{2} \int i (a+b \text {arcsinh}(c+d x))^3 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \int (a+b \text {arcsinh}(c+d x)) \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)\right )-\frac {1}{2} i \int (a+b \text {arcsinh}(c+d x))^3 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (i b \int \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-i b \int \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )\right )-\frac {1}{2} i \left (3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {1}{2} i \left (3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (-i b \int e^{-\text {arcsinh}(c+d x)} \log \left (1+e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}+i b \int e^{-\text {arcsinh}(c+d x)} \log (-c-d x+1)de^{\text {arcsinh}(c+d x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {1}{2} i \left (3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {1}{2} i \left (-3 i b \left (2 b \int (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+3 i b \left (2 b \int (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {1}{2} i \left (-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (-\frac {1}{2} i \left (3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}(3,-c-d x)de^{\text {arcsinh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )+\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{3 (c+d x)^3}+\frac {4}{3} b \left (\frac {3}{2} b \left (-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )\right )-\frac {1}{2} i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \operatorname {PolyLog}(4,-c-d x)\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{2 (c+d x)^2}\right )}{d e^4}\) |
(-1/3*(a + b*ArcSinh[c + d*x])^4/(c + d*x)^3 + (4*b*(-1/2*(Sqrt[1 + (c + d *x)^2]*(a + b*ArcSinh[c + d*x])^3)/(c + d*x)^2 + (3*b*(-((a + b*ArcSinh[c + d*x])^2/(c + d*x)) + (2*I)*b*((2*I)*(a + b*ArcSinh[c + d*x])*ArcTanh[E^A rcSinh[c + d*x]] - I*b*PolyLog[2, E^ArcSinh[c + d*x]] + I*b*PolyLog[2, -c - d*x])))/2 - (I/2)*((2*I)*(a + b*ArcSinh[c + d*x])^3*ArcTanh[E^ArcSinh[c + d*x]] + (3*I)*b*(-((a + b*ArcSinh[c + d*x])^2*PolyLog[2, E^ArcSinh[c + d *x]]) + 2*b*((a + b*ArcSinh[c + d*x])*PolyLog[3, E^ArcSinh[c + d*x]] - b*P olyLog[4, E^ArcSinh[c + d*x]])) - (3*I)*b*(-((a + b*ArcSinh[c + d*x])^2*Po lyLog[2, -E^ArcSinh[c + d*x]]) + 2*b*((a + b*ArcSinh[c + d*x])*PolyLog[3, -E^ArcSinh[c + d*x]] - b*PolyLog[4, -c - d*x])))))/3)/(d*e^4)
3.2.54.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Sim p[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Int[(((a_.) + ArcSinh[(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*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_.) + ArcSinh[(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* ArcSinh[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 = 0.56 (sec) , antiderivative size = 883, normalized size of antiderivative = 2.29
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(883\) |
| default | \(\text {Expression too large to display}\) | \(883\) |
| parts | \(\text {Expression too large to display}\) | \(894\) |
1/d*(-1/3*a^4/e^4/(d*x+c)^3+b^4/e^4*(-1/3/(d*x+c)^3*arcsinh(d*x+c)^2*(2*(1 +(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)+arcsinh(d*x+c)^2+6*(d*x+c)^2)+2/3 *arcsinh(d*x+c)^3*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)^2*polyl og(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-4*arcsinh(d*x+c)*polylog(3,-d*x-c-(1+(d*x +c)^2)^(1/2))+4*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))-2/3*arcsinh(d*x+c)^3 *ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))-2*arcsinh(d*x+c)^2*polylog(2,d*x+c+(1+(d* x+c)^2)^(1/2))+4*arcsinh(d*x+c)*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))-4*pol ylog(4,d*x+c+(1+(d*x+c)^2)^(1/2))-4*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2 )^(1/2))-4*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+4*arcsinh(d*x+c)*ln(1-d*x -c-(1+(d*x+c)^2)^(1/2))+4*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*a*b^3/e^ 4*(-1/6/(d*x+c)^3*arcsinh(d*x+c)*(3*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d* x+c)+2*arcsinh(d*x+c)^2+6*(d*x+c)^2)+1/2*arcsinh(d*x+c)^2*ln(1+d*x+c+(1+(d *x+c)^2)^(1/2))+arcsinh(d*x+c)*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-polyl og(3,-d*x-c-(1+(d*x+c)^2)^(1/2))-1/2*arcsinh(d*x+c)^2*ln(1-d*x-c-(1+(d*x+c )^2)^(1/2))-arcsinh(d*x+c)*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))+polylog(3, d*x+c+(1+(d*x+c)^2)^(1/2))-2*arctanh(d*x+c+(1+(d*x+c)^2)^(1/2)))+6*a^2*b^2 /e^4*(-1/3*((1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)+arcsinh(d*x+c)^2+(d *x+c)^2)/(d*x+c)^3+1/3*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+1/3* polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-1/3*arcsinh(d*x+c)*ln(1-d*x-c-(1+(d* x+c)^2)^(1/2))-1/3*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*b*a^3/e^4*(-...
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2* arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh(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)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{4}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(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*asinh(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*asinh(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*asinh(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*asinh(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
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*b^4*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^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*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) + integrate(2/3*(2*(3*(c^3 + c)*a*b^3 + (c^3 + c)*b^4 + (3*a*b^3*d^3 + b^4*d^3)*x^3 + 3*(3*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (3*(3*c^2*d + d)*a*b^3 + (3*c^2*d + d)*b^4)*x + (b^4*c ^2 + 3*(c^2 + 1)*a*b^3 + (3*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(3*a*b^3*c*d + b^ 4*c*d)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + 9*(a^2*b^2*d^3*x^3 + 3*a^2*b^2*c*d^2*x^2 + (3*c^2* d + d)*a^2*b^2*x + (c^3 + c)*a^2*b^2 + (a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + (c^2 + 1)*a^2*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt (d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 6*(a^3*b*d^3*x^3 + 3*a^3*b*c*d^2*x^2 + (3*c^2*d + d)*a^3*b*x + (c^3 + c)*a^3*b + (a^3*b*d^2*x^2 + 2*a^3*b*c*d*x + (c^2 + 1)*a^3*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^ 2*x^2 + 2*c*d*x + c^2 + 1)))/(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4 + c^ 5*e^4 + (21*c^2*d^5*e^4 + d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 + c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 + 2*c^2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 + 10*c^3*d^2*e^ 4)*x^2 + (7*c^6*d*e^4 + 5*c^4*d*e^4)*x + (d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 + c^4*e^4 + (15*c^2*d^4*e^4 + d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 + c* d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^4 + 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 + 2* c^3*d*e^4)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]