\(\int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [229]

Optimal result

Integrand size = 24, antiderivative size = 461 \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {3 a \sqrt {1-a x} \text {arccosh}(a x)^2}{2 x \sqrt {-1+a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}+\frac {6 a^2 \sqrt {1-a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {a^2 \sqrt {1-a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {3 i a^2 \sqrt {1-a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {3 i a^2 \sqrt {1-a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-1+a x}}+\frac {3 i a^2 \sqrt {1-a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {3 i a^2 \sqrt {1-a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-1+a x}}-\frac {3 i a^2 \sqrt {1-a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {3 i a^2 \sqrt {1-a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {3 i a^2 \sqrt {1-a x} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {3 i a^2 \sqrt {1-a x} \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}} \] Output:

-3/2*a*(-a*x+1)^(1/2)*arccosh(a*x)^2/x/(a*x-1)^(1/2)-1/2*(-a^2*x^2+1)^(1/2 
)*arccosh(a*x)^3/x^2+6*a^2*(-a*x+1)^(1/2)*arccosh(a*x)*arctan(a*x+(a*x-1)^ 
(1/2)*(a*x+1)^(1/2))/(a*x-1)^(1/2)-a^2*(-a*x+1)^(1/2)*arccosh(a*x)^3*arcta 
n(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a*x-1)^(1/2)-3*I*a^2*(-a*x+1)^(1/2)*po 
lylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)+3/2*I*a^2*(-a* 
x+1)^(1/2)*arccosh(a*x)^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/ 
(a*x-1)^(1/2)+3*I*a^2*(-a*x+1)^(1/2)*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1 
)^(1/2)))/(a*x-1)^(1/2)-3/2*I*a^2*(-a*x+1)^(1/2)*arccosh(a*x)^2*polylog(2, 
I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)-3*I*a^2*(-a*x+1)^(1/2)* 
arccosh(a*x)*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2) 
+3*I*a^2*(-a*x+1)^(1/2)*arccosh(a*x)*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1 
)^(1/2)))/(a*x-1)^(1/2)+3*I*a^2*(-a*x+1)^(1/2)*polylog(4,-I*(a*x+(a*x-1)^( 
1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)-3*I*a^2*(-a*x+1)^(1/2)*polylog(4,I*(a*x 
+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1051\) vs. \(2(461)=922\).

Time = 3.95 (sec) , antiderivative size = 1051, normalized size of antiderivative = 2.28 \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx =\text {Too large to display} \] Input:

Integrate[ArcCosh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]
 

Output:

((-1/128*I)*a^2*(1 + a*x)*(7*Pi^4*Sqrt[(-1 + a*x)/(1 + a*x)] + (8*I)*Pi^3* 
Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + 24*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x 
)]*ArcCosh[a*x]^2 + ((192*I)*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2)/(a 
*x) + ((64*I)*(-1 + a*x)*ArcCosh[a*x]^3)/(a^2*x^2) - (32*I)*Pi*Sqrt[(-1 + 
a*x)/(1 + a*x)]*ArcCosh[a*x]^3 - 16*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x 
]^4 - 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x] 
] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I/E^ArcCosh[a*x]] + 384* 
Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^ 
2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] - (96* 
I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] 
- 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 
 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I*E^ArcCosh[a*x]] 
 + (96*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh 
[a*x]] - (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I*E^ArcCosh[a*x]] + 
 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] + 
(8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4] 
] - 48*Sqrt[(-1 + a*x)/(1 + a*x)]*(8 + Pi^2 - (4*I)*Pi*ArcCosh[a*x] - 4*Ar 
cCosh[a*x]^2)*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a 
*x)]*PolyLog[2, I/E^ArcCosh[a*x]] + 192*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh 
[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + ...
 

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.59, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6347, 6298, 6361, 3042, 4668, 3011, 6362, 3042, 4668, 2715, 2838, 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.

\(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[ArcCosh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]
 

Output:

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

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 6347

Int[((a_.) + ArcCosh[(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*ArcCosh[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*ArcCosh[c*x])^n, x], x] + Simp 
[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*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*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

rule 6361

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x 
]/Sqrt[d + e*x^2])]   Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] 
, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int 
egerQ[m]
 

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]
 

Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{3} \sqrt {-a^{2} x^{2}+1}}d x\]

Input:

int(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3/(a^2*x^5 - x^3), x)
 

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(acosh(a*x)**3/x**3/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(acosh(a*x)**3/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(arccosh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^3), x)
 

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(arccosh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(acosh(a*x)^3/(x^3*(1 - a^2*x^2)^(1/2)),x)
 

Output:

int(acosh(a*x)^3/(x^3*(1 - a^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x \] Input:

int(acosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(acosh(a*x)**3/(sqrt( - a**2*x**2 + 1)*x**3),x)