Integrand size = 23, antiderivative size = 261 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \]
-b^2*(a+b*arcsinh(d*x+c))/d/e^4/(d*x+c)-1/3*(a+b*arcsinh(d*x+c))^3/d/e^4/( d*x+c)^3+b*(a+b*arcsinh(d*x+c))^2*arctanh(d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4 -b^3*arctanh((1+(d*x+c)^2)^(1/2))/d/e^4+b^2*(a+b*arcsinh(d*x+c))*polylog(2 ,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4-b^2*(a+b*arcsinh(d*x+c))*polylog(2,d*x+ c+(1+(d*x+c)^2)^(1/2))/d/e^4-b^3*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e ^4+b^3*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4-1/2*b*(a+b*arcsinh(d*x+c ))^2*(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. \(694\) vs. \(2(261)=522\).
Time = 7.25 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.66 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {a^3}{3 d e^4 (c+d x)^3}-\frac {a^2 b \sqrt {1+c^2+2 c d x+d^2 x^2}}{2 d e^4 (c+d x)^2}-\frac {a^2 b \text {arcsinh}(c+d x)}{d e^4 (c+d x)^3}-\frac {a^2 b \log (c+d x)}{2 d e^4}+\frac {a^2 b \log \left (1+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )}{2 d e^4}+\frac {a b^2 \left (-8 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-\frac {2 \left (-2+4 \text {arcsinh}(c+d x)^2+2 \cosh (2 \text {arcsinh}(c+d x))-3 (c+d x) \text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )+3 (c+d x) \text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )-4 (c+d x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+2 \text {arcsinh}(c+d x) \sinh (2 \text {arcsinh}(c+d x))+\text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right ) \sinh (3 \text {arcsinh}(c+d x))-\text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right ) \sinh (3 \text {arcsinh}(c+d x))\right )}{(c+d x)^3}\right )}{8 d e^4}+\frac {b^3 \left (-24 \text {arcsinh}(c+d x) \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )+4 \text {arcsinh}(c+d x)^3 \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-6 \text {arcsinh}(c+d x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-(c+d x) \text {arcsinh}(c+d x)^3 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-24 \text {arcsinh}(c+d x)^2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+48 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )+48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c+d x)}\right )-6 \text {arcsinh}(c+d x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {16 \text {arcsinh}(c+d x)^3 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )}{(c+d x)^3}+24 \text {arcsinh}(c+d x) \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-4 \text {arcsinh}(c+d x)^3 \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )}{48 d e^4} \]
-1/3*a^3/(d*e^4*(c + d*x)^3) - (a^2*b*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2])/( 2*d*e^4*(c + d*x)^2) - (a^2*b*ArcSinh[c + d*x])/(d*e^4*(c + d*x)^3) - (a^2 *b*Log[c + d*x])/(2*d*e^4) + (a^2*b*Log[1 + Sqrt[1 + c^2 + 2*c*d*x + d^2*x ^2]])/(2*d*e^4) + (a*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)*ArcSinh[c + d*x]*Log[ 1 + E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2, E^(-ArcSinh[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))/(8*d*e^ 4) + (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*Csch[ArcSinh[c + d*x ]/2]^2 - (c + d*x)*ArcSinh[c + d*x]^3*Csch[ArcSinh[c + d*x]/2]^4 - 24*ArcS inh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] + 24*ArcSinh[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*Po lyLog[3, E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]^2*Sech[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*Ta...
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.85, 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, 243, 73, 220, 6231, 3042, 26, 4670, 3011, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^3}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {b \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x)^3 \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {b \left (b \int \frac {a+b \text {arcsinh}(c+d x)}{(c+d x)^2}d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {b \left (b \left (b \int \frac {1}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {b \left (b \left (\frac {1}{2} b \int \frac {1}{(c+d x)^2 \sqrt {(c+d x)^2+1}}d(c+d x)^2-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {b \left (b \left (b \int \frac {1}{(c+d x)^4-1}d\sqrt {(c+d x)^2+1}-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {b \left (-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {b \left (-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}d\text {arcsinh}(c+d x)+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}+b \left (-\frac {1}{2} \int i (a+b \text {arcsinh}(c+d x))^2 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}+b \left (-\frac {1}{2} i \int (a+b \text {arcsinh}(c+d x))^2 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}+b \left (-\frac {1}{2} i \left (2 i b \int (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-2 i b \int (a+b \text {arcsinh}(c+d x)) \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}+b \left (-\frac {1}{2} i \left (-2 i b \left (b \int \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )+2 i b \left (b \int \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}+b \left (-\frac {1}{2} i \left (2 i b \left (b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}(2,-c-d x)de^{\text {arcsinh}(c+d x)}-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 (c+d x)^3}+b \left (-\frac {1}{2} i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )-2 i b \left (b \operatorname {PolyLog}(3,-c-d x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )\right )+b \left (-\frac {a+b \text {arcsinh}(c+d x)}{c+d x}-b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 (c+d x)^2}\right )}{d e^4}\) |
(-1/3*(a + b*ArcSinh[c + d*x])^3/(c + d*x)^3 + b*(-1/2*(Sqrt[1 + (c + d*x) ^2]*(a + b*ArcSinh[c + d*x])^2)/(c + d*x)^2 + b*(-((a + b*ArcSinh[c + d*x] )/(c + d*x)) - b*ArcTanh[Sqrt[1 + (c + d*x)^2]]) - (I/2)*((2*I)*(a + b*Arc Sinh[c + d*x])^2*ArcTanh[E^ArcSinh[c + d*x]] + (2*I)*b*(-((a + b*ArcSinh[c + d*x])*PolyLog[2, E^ArcSinh[c + d*x]]) + b*PolyLog[3, E^ArcSinh[c + d*x] ]) - (2*I)*b*(-((a + b*ArcSinh[c + d*x])*PolyLog[2, -E^ArcSinh[c + d*x]]) + b*PolyLog[3, -c - d*x]))))/(d*e^4)
3.2.45.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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[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]
Time = 0.57 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.86
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+2 \operatorname {arcsinh}\left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}+\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{4}}+\frac {3 a \,b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(486\) |
| default | \(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+2 \operatorname {arcsinh}\left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}+\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{4}}+\frac {3 a \,b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(486\) |
| parts | \(-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+2 \operatorname {arcsinh}\left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}+\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{4} d}+\frac {3 a \,b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4} d}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4} d}\) | \(494\) |
1/d*(-1/3*a^3/e^4/(d*x+c)^3+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))-polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))-1/2*arc sinh(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)))+3*a*b^2/e^4*(-1/3*((1+(d*x+c)^2)^(1/2)*(d*x+c)*ar csinh(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)))+3*a^2*b/e^4*(-1/3/(d*x+c)^3*arcsinh(d*x+c)-1/6/(d*x+c)^2*(1 +(d*x+c)^2)^(1/2)+1/6*arctanh(1/(1+(d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*ar csinh(d*x + c) + a^3)/(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)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(Integral(a**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(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(3*a*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(3*a**2*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
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*b^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^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) - 1/3*a^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(((3*(c^3 + c)*a* b^2 + (c^3 + c)*b^3 + (3*a*b^2*d^3 + b^3*d^3)*x^3 + 3*(3*a*b^2*c*d^2 + b^3 *c*d^2)*x^2 + (3*(3*c^2*d + d)*a*b^2 + (3*c^2*d + d)*b^3)*x + (b^3*c^2 + 3 *(c^2 + 1)*a*b^2 + (3*a*b^2*d^2 + b^3*d^2)*x^2 + 2*(3*a*b^2*c*d + b^3*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))^2 + 3*(a^2*b*d^3*x^3 + 3*a^2*b*c*d^2*x^2 + (3*c^2*d + d)*a^2 *b*x + (c^3 + c)*a^2*b + (a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + (c^2 + 1)*a^2*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)*sqr t(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]