3.1.41 \(\int \frac {\text {arccosh}(a x)^4}{x^4} \, dx\) [41]

3.1.41.1 Optimal result

Integrand size = 10, antiderivative size = 268 \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\frac {2 a^2 \text {arccosh}(a x)^2}{x}+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 x^2}-\frac {\text {arccosh}(a x)^4}{3 x^3}-8 a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )+\frac {4}{3} a^3 \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+2 i a^3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-4 i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right ) \]

output

2*a^2*arccosh(a*x)^2/x-1/3*arccosh(a*x)^4/x^3-8*a^3*arccosh(a*x)*arctan(a* 
x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+4/3*a^3*arccosh(a*x)^3*arctan(a*x+(a*x-1)^( 
1/2)*(a*x+1)^(1/2))+4*I*a^3*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)) 
)-2*I*a^3*arccosh(a*x)^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-4 
*I*a^3*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+2*I*a^3*arccosh(a*x) 
^2*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+4*I*a^3*arccosh(a*x)*pol 
ylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-4*I*a^3*arccosh(a*x)*polylog( 
3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-4*I*a^3*polylog(4,-I*(a*x+(a*x-1)^( 
1/2)*(a*x+1)^(1/2)))+4*I*a^3*polylog(4,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)) 
)+2/3*a*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^2
 

3.1.41.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(595\) vs. \(2(268)=536\).

Time = 2.11 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.22 \[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=a^3 \left (\frac {1}{2} i \left (8+\pi ^2-4 i \pi \text {arccosh}(a x)-4 \text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-\frac {1}{96} i \left (7 \pi ^4+8 i \pi ^3 \text {arccosh}(a x)+24 \pi ^2 \text {arccosh}(a x)^2+\frac {192 i \text {arccosh}(a x)^2}{a x}-32 i \pi \text {arccosh}(a x)^3+\frac {64 i \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^3}{a^2 x^2}-16 \text {arccosh}(a x)^4-\frac {32 i \text {arccosh}(a x)^4}{a^3 x^3}-384 \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+384 \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )+48 \pi ^2 \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-96 i \pi \text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-64 \text {arccosh}(a x)^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-48 \pi ^2 \text {arccosh}(a x) \log \left (1-i e^{\text {arccosh}(a x)}\right )+96 i \pi \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+64 \text {arccosh}(a x)^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arccosh}(a x))\right )\right )+384 \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+192 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-48 \pi ^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+384 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-384 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-192 i \pi \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{-\text {arccosh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )\right ) \]

input

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

output

a^3*((I/2)*(8 + Pi^2 - (4*I)*Pi*ArcCosh[a*x] - 4*ArcCosh[a*x]^2)*PolyLog[2 
, (-I)/E^ArcCosh[a*x]] - (I/96)*(7*Pi^4 + (8*I)*Pi^3*ArcCosh[a*x] + 24*Pi^ 
2*ArcCosh[a*x]^2 + ((192*I)*ArcCosh[a*x]^2)/(a*x) - (32*I)*Pi*ArcCosh[a*x] 
^3 + ((64*I)*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^3)/(a^2*x^2 
) - 16*ArcCosh[a*x]^4 - ((32*I)*ArcCosh[a*x]^4)/(a^3*x^3) - 384*ArcCosh[a* 
x]*Log[1 - I/E^ArcCosh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcCosh[a*x]] + 384* 
ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*ArcCosh[a*x]*Log[1 + I/E^ 
ArcCosh[a*x]] - (96*I)*Pi*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*Ar 
cCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*ArcCosh[a*x]*Log[1 - I*E^ 
ArcCosh[a*x]] + (96*I)*Pi*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I) 
*Pi^3*Log[1 + I*E^ArcCosh[a*x]] + 64*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a* 
x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4]] + 384*PolyLog[2, I/ 
E^ArcCosh[a*x]] + 192*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 48* 
Pi^2*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*ArcCosh[a*x]*PolyLog[2, I*E 
^ArcCosh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcCosh[a*x]] + 384*ArcCosh[ 
a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 384*ArcCosh[a*x]*PolyLog[3, (-I)*E^ 
ArcCosh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcCosh[a*x]] + 384*PolyLog[4, ( 
-I)/E^ArcCosh[a*x]] + 384*PolyLog[4, (-I)*E^ArcCosh[a*x]]))
 

3.1.41.3 Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6298, 6348, 6298, 6362, 3042, 4668, 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.

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6348

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

-1/3*ArcCosh[a*x]^4/x^3 + (4*a*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] 
^3)/(2*x^2) - (3*a*(-(ArcCosh[a*x]^2/x) + 2*a*(2*ArcCosh[a*x]*ArcTan[E^Arc 
Cosh[a*x]] - I*PolyLog[2, (-I)*E^ArcCosh[a*x]] + I*PolyLog[2, I*E^ArcCosh[ 
a*x]])))/2 + (a^2*(2*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] + (3*I)*(-(ArcC 
osh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, ( 
-I)*E^ArcCosh[a*x]] - PolyLog[4, (-I)*E^ArcCosh[a*x]])) - (3*I)*(-(ArcCosh 
[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, I*E^Arc 
Cosh[a*x]] - PolyLog[4, I*E^ArcCosh[a*x]]))))/2))/3
 

3.1.41.3.1 Defintions of rubi rules used

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 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

rule 6348

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) 
*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( 
m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)* 
(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f 
*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] 
   Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos 
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq 
Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 6362

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 
_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ 
Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Subst 
[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, 
 e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte 
gerQ[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]
 

3.1.41.4 Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{4}}{x^{4}}d x\]

input

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

output

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

3.1.41.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \]

input

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

output

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

3.1.41.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{4}}\, dx \]

input

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

output

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

3.1.41.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \]

input

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

output

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

3.1.41.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{4}} \,d x } \]

input

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

output

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

3.1.41.9 Mupad [F(-1)]

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

input

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

output

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