\(\int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 223 \[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=-\frac {2 a^2 \text {arcsinh}(a x)^2}{x}-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 x^2}-\frac {\text {arcsinh}(a x)^4}{3 x^3}-8 a^3 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {4}{3} a^3 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-4 a^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+2 a^3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+4 a^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-2 a^3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-4 a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+4 a^3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )+4 a^3 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )-4 a^3 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \] Output:

-2*a^2*arcsinh(a*x)^2/x-2/3*a*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^3/x^2-1/3*arc 
sinh(a*x)^4/x^3-8*a^3*arcsinh(a*x)*arctanh(a*x+(a^2*x^2+1)^(1/2))+4/3*a^3* 
arcsinh(a*x)^3*arctanh(a*x+(a^2*x^2+1)^(1/2))-4*a^3*polylog(2,-a*x-(a^2*x^ 
2+1)^(1/2))+2*a^3*arcsinh(a*x)^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))+4*a^3*p 
olylog(2,a*x+(a^2*x^2+1)^(1/2))-2*a^3*arcsinh(a*x)^2*polylog(2,a*x+(a^2*x^ 
2+1)^(1/2))-4*a^3*arcsinh(a*x)*polylog(3,-a*x-(a^2*x^2+1)^(1/2))+4*a^3*arc 
sinh(a*x)*polylog(3,a*x+(a^2*x^2+1)^(1/2))+4*a^3*polylog(4,-a*x-(a^2*x^2+1 
)^(1/2))-4*a^3*polylog(4,a*x+(a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.59 \[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\frac {1}{24} a^3 \left (-2 \pi ^4+4 \text {arcsinh}(a x)^4-24 \text {arcsinh}(a x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )+2 \text {arcsinh}(a x)^4 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-\frac {1}{2} a x \text {arcsinh}(a x)^4 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(a x)\right )+96 \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )-96 \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )+16 \text {arcsinh}(a x)^3 \log \left (1+e^{-\text {arcsinh}(a x)}\right )-16 \text {arcsinh}(a x)^3 \log \left (1-e^{\text {arcsinh}(a x)}\right )-48 \left (-2+\text {arcsinh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-96 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-48 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-96 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )+96 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-96 \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(a x)}\right )-96 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right )-4 \text {arcsinh}(a x)^3 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-\frac {8 \text {arcsinh}(a x)^4 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(a x)\right )}{a^3 x^3}+24 \text {arcsinh}(a x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )-2 \text {arcsinh}(a x)^4 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right ) \] Input:

Integrate[ArcSinh[a*x]^4/x^4,x]
 

Output:

(a^3*(-2*Pi^4 + 4*ArcSinh[a*x]^4 - 24*ArcSinh[a*x]^2*Coth[ArcSinh[a*x]/2] 
+ 2*ArcSinh[a*x]^4*Coth[ArcSinh[a*x]/2] - 4*ArcSinh[a*x]^3*Csch[ArcSinh[a* 
x]/2]^2 - (a*x*ArcSinh[a*x]^4*Csch[ArcSinh[a*x]/2]^4)/2 + 96*ArcSinh[a*x]* 
Log[1 - E^(-ArcSinh[a*x])] - 96*ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a*x])] + 
16*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] - 16*ArcSinh[a*x]^3*Log[1 - E 
^ArcSinh[a*x]] - 48*(-2 + ArcSinh[a*x]^2)*PolyLog[2, -E^(-ArcSinh[a*x])] - 
 96*PolyLog[2, E^(-ArcSinh[a*x])] - 48*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh 
[a*x]] - 96*ArcSinh[a*x]*PolyLog[3, -E^(-ArcSinh[a*x])] + 96*ArcSinh[a*x]* 
PolyLog[3, E^ArcSinh[a*x]] - 96*PolyLog[4, -E^(-ArcSinh[a*x])] - 96*PolyLo 
g[4, E^ArcSinh[a*x]] - 4*ArcSinh[a*x]^3*Sech[ArcSinh[a*x]/2]^2 - (8*ArcSin 
h[a*x]^4*Sinh[ArcSinh[a*x]/2]^4)/(a^3*x^3) + 24*ArcSinh[a*x]^2*Tanh[ArcSin 
h[a*x]/2] - 2*ArcSinh[a*x]^4*Tanh[ArcSinh[a*x]/2]))/24
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.15 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {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.

\(\displaystyle \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {4}{3} a \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {a^2 x^2+1}}dx-\frac {\text {arcsinh}(a x)^4}{3 x^3}\)

\(\Big \downarrow \) 6224

\(\displaystyle \frac {4}{3} a \left (-\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {a^2 x^2+1}}dx+\frac {3}{2} a \int \frac {\text {arcsinh}(a x)^2}{x^2}dx-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )-\frac {\text {arcsinh}(a x)^4}{3 x^3}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {4}{3} a \left (\frac {3}{2} a \left (2 a \int \frac {\text {arcsinh}(a x)}{x \sqrt {a^2 x^2+1}}dx-\frac {\text {arcsinh}(a x)^2}{x}\right )-\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {a^2 x^2+1}}dx-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )-\frac {\text {arcsinh}(a x)^4}{3 x^3}\)

\(\Big \downarrow \) 6231

\(\displaystyle \frac {4}{3} a \left (-\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)^3}{a x}d\text {arcsinh}(a x)+\frac {3}{2} a \left (2 a \int \frac {\text {arcsinh}(a x)}{a x}d\text {arcsinh}(a x)-\frac {\text {arcsinh}(a x)^2}{x}\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )-\frac {\text {arcsinh}(a x)^4}{3 x^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} a^2 \int i \text {arcsinh}(a x)^3 \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 a \int i \text {arcsinh}(a x) \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \int \text {arcsinh}(a x)^3 \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \int \text {arcsinh}(a x) \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (3 i \int \text {arcsinh}(a x)^2 \log \left (1-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-3 i \int \text {arcsinh}(a x)^2 \log \left (1+e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (i \int \log \left (1-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-i \int \log \left (1+e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (3 i \int \text {arcsinh}(a x)^2 \log \left (1-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-3 i \int \text {arcsinh}(a x)^2 \log \left (1+e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (i \int e^{-\text {arcsinh}(a x)} \log \left (1-e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}-i \int e^{-\text {arcsinh}(a x)} \log \left (1+e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}+2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (3 i \int \text {arcsinh}(a x)^2 \log \left (1-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-3 i \int \text {arcsinh}(a x)^2 \log \left (1+e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (-3 i \left (2 \int \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )\right )+3 i \left (2 \int \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )+2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\right )\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (-3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )\right )+3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )+2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\right )\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (-3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-\int e^{-\text {arcsinh}(a x)} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )\right )+3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\int e^{-\text {arcsinh}(a x)} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )+2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {\text {arcsinh}(a x)^4}{3 x^3}+\frac {4}{3} a \left (-\frac {1}{2} i a^2 \left (2 i \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )\right )+3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right )\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^2}{x}+2 i a \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\right )\right )\)

Input:

Int[ArcSinh[a*x]^4/x^4,x]
 

Output:

-1/3*ArcSinh[a*x]^4/x^3 + (4*a*(-1/2*(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/x^ 
2 + (3*a*(-(ArcSinh[a*x]^2/x) + (2*I)*a*((2*I)*ArcSinh[a*x]*ArcTanh[E^ArcS 
inh[a*x]] + I*PolyLog[2, -E^ArcSinh[a*x]] - I*PolyLog[2, E^ArcSinh[a*x]])) 
)/2 - (I/2)*a^2*((2*I)*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - (3*I)*(-(A 
rcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, -E 
^ArcSinh[a*x]] - PolyLog[4, -E^ArcSinh[a*x]])) + (3*I)*(-(ArcSinh[a*x]^2*P 
olyLog[2, E^ArcSinh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - 
PolyLog[4, E^ArcSinh[a*x]])))))/3
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2715

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]
 

rule 2720

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]]]
 

rule 2838

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]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

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]
 

rule 6191

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]
 

rule 6224

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]
 

rule 6231

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]
 

rule 7143

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]
 

rule 7163

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]
 

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\operatorname {arcsinh}\left (x a \right )^{2} \left (2 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a +\operatorname {arcsinh}\left (x a \right )^{2}+6 a^{2} x^{2}\right )}{3 x^{3} a^{3}}+\frac {2 \operatorname {arcsinh}\left (x a \right )^{3} \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )}{3}+2 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (3, -x a -\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (4, -x a -\sqrt {a^{2} x^{2}+1}\right )-\frac {2 \operatorname {arcsinh}\left (x a \right )^{3} \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )}{3}-2 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (3, x a +\sqrt {a^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (4, x a +\sqrt {a^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x a \right ) \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (x a \right ) \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(340\)
default	\(a^{3} \left (-\frac {\operatorname {arcsinh}\left (x a \right )^{2} \left (2 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a +\operatorname {arcsinh}\left (x a \right )^{2}+6 a^{2} x^{2}\right )}{3 x^{3} a^{3}}+\frac {2 \operatorname {arcsinh}\left (x a \right )^{3} \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )}{3}+2 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (3, -x a -\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (4, -x a -\sqrt {a^{2} x^{2}+1}\right )-\frac {2 \operatorname {arcsinh}\left (x a \right )^{3} \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )}{3}-2 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (3, x a +\sqrt {a^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (4, x a +\sqrt {a^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x a \right ) \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (x a \right ) \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )\right )\)	\(340\)

Input:

int(arcsinh(x*a)^4/x^4,x,method=_RETURNVERBOSE)
 

Output:

a^3*(-1/3/x^3/a^3*arcsinh(x*a)^2*(2*arcsinh(x*a)*(a^2*x^2+1)^(1/2)*x*a+arc 
sinh(x*a)^2+6*a^2*x^2)+2/3*arcsinh(x*a)^3*ln(1+x*a+(a^2*x^2+1)^(1/2))+2*ar 
csinh(x*a)^2*polylog(2,-x*a-(a^2*x^2+1)^(1/2))-4*arcsinh(x*a)*polylog(3,-x 
*a-(a^2*x^2+1)^(1/2))+4*polylog(4,-x*a-(a^2*x^2+1)^(1/2))-2/3*arcsinh(x*a) 
^3*ln(1-x*a-(a^2*x^2+1)^(1/2))-2*arcsinh(x*a)^2*polylog(2,x*a+(a^2*x^2+1)^ 
(1/2))+4*arcsinh(x*a)*polylog(3,x*a+(a^2*x^2+1)^(1/2))-4*polylog(4,x*a+(a^ 
2*x^2+1)^(1/2))-4*arcsinh(x*a)*ln(1+x*a+(a^2*x^2+1)^(1/2))-4*polylog(2,-x* 
a-(a^2*x^2+1)^(1/2))+4*arcsinh(x*a)*ln(1-x*a-(a^2*x^2+1)^(1/2))+4*polylog( 
2,x*a+(a^2*x^2+1)^(1/2)))
 

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{4}} \,d x } \] Input:

integrate(arcsinh(a*x)^4/x^4,x, algorithm="fricas")
 

Output:

integral(arcsinh(a*x)^4/x^4, x)
 

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\int \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{x^{4}}\, dx \] Input:

integrate(asinh(a*x)**4/x**4,x)
 

Output:

Integral(asinh(a*x)**4/x**4, x)
 

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{4}} \,d x } \] Input:

integrate(arcsinh(a*x)^4/x^4,x, algorithm="maxima")
 

Output:

-1/3*log(a*x + sqrt(a^2*x^2 + 1))^4/x^3 + integrate(4/3*(a^3*x^2 + sqrt(a^ 
2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2 + 1))^3/(a^3*x^6 + a*x^4 + (a 
^2*x^5 + x^3)*sqrt(a^2*x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{4}} \,d x } \] Input:

integrate(arcsinh(a*x)^4/x^4,x, algorithm="giac")
 

Output:

integrate(arcsinh(a*x)^4/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{x^4} \,d x \] Input:

int(asinh(a*x)^4/x^4,x)
 

Output:

int(asinh(a*x)^4/x^4, x)
 

Reduce [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x^4} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{4}}{x^{4}}d x \] Input:

int(asinh(a*x)^4/x^4,x)
 

Output:

int(asinh(a*x)**4/x**4,x)