3.4.32 \(\int \frac {\text {arcsinh}(a x)^3}{(c+a^2 c x^2)^2} \, dx\) [332]

3.4.32.1 Optimal result

Integrand size = 19, antiderivative size = 294 \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3 \text {arcsinh}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2}}+\frac {x \text {arcsinh}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {6 \text {arcsinh}(a x) \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c^2}+\frac {\text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c^2}+\frac {3 i \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c^2}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{2 a c^2}-\frac {3 i \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c^2}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{2 a c^2}+\frac {3 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c^2}-\frac {3 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c^2}-\frac {3 i \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )}{a c^2}+\frac {3 i \operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )}{a c^2} \]

output

1/2*x*arcsinh(a*x)^3/c^2/(a^2*x^2+1)-6*arcsinh(a*x)*arctan(a*x+(a^2*x^2+1) 
^(1/2))/a/c^2+arcsinh(a*x)^3*arctan(a*x+(a^2*x^2+1)^(1/2))/a/c^2+3*I*polyl 
og(2,-I*(a*x+(a^2*x^2+1)^(1/2)))/a/c^2-3/2*I*arcsinh(a*x)^2*polylog(2,-I*( 
a*x+(a^2*x^2+1)^(1/2)))/a/c^2-3*I*polylog(2,I*(a*x+(a^2*x^2+1)^(1/2)))/a/c 
^2+3/2*I*arcsinh(a*x)^2*polylog(2,I*(a*x+(a^2*x^2+1)^(1/2)))/a/c^2+3*I*arc 
sinh(a*x)*polylog(3,-I*(a*x+(a^2*x^2+1)^(1/2)))/a/c^2-3*I*arcsinh(a*x)*pol 
ylog(3,I*(a*x+(a^2*x^2+1)^(1/2)))/a/c^2-3*I*polylog(4,-I*(a*x+(a^2*x^2+1)^ 
(1/2)))/a/c^2+3*I*polylog(4,I*(a*x+(a^2*x^2+1)^(1/2)))/a/c^2+3/2*arcsinh(a 
*x)^2/a/c^2/(a^2*x^2+1)^(1/2)
 

3.4.32.2 Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.93 \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {i \left (7 \pi ^4+8 i \pi ^3 \text {arcsinh}(a x)+24 \pi ^2 \text {arcsinh}(a x)^2+\frac {192 i \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}}-32 i \pi \text {arcsinh}(a x)^3+\frac {64 i a x \text {arcsinh}(a x)^3}{1+a^2 x^2}-16 \text {arcsinh}(a x)^4-384 \text {arcsinh}(a x) \log \left (1-i e^{-\text {arcsinh}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\text {arcsinh}(a x)}\right )+384 \text {arcsinh}(a x) \log \left (1+i e^{-\text {arcsinh}(a x)}\right )+48 \pi ^2 \text {arcsinh}(a x) \log \left (1+i e^{-\text {arcsinh}(a x)}\right )-96 i \pi \text {arcsinh}(a x)^2 \log \left (1+i e^{-\text {arcsinh}(a x)}\right )-64 \text {arcsinh}(a x)^3 \log \left (1+i e^{-\text {arcsinh}(a x)}\right )-48 \pi ^2 \text {arcsinh}(a x) \log \left (1-i e^{\text {arcsinh}(a x)}\right )+96 i \pi \text {arcsinh}(a x)^2 \log \left (1-i e^{\text {arcsinh}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\text {arcsinh}(a x)}\right )+64 \text {arcsinh}(a x)^3 \log \left (1+i e^{\text {arcsinh}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a x))\right )\right )-48 \left (8+\pi ^2-4 i \pi \text {arcsinh}(a x)-4 \text {arcsinh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(a x)}\right )+384 \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(a x)}\right )+192 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )-48 \pi ^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )+192 i \pi \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )+192 i \pi \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(a x)}\right )+384 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(a x)}\right )-384 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )-192 i \pi \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{-\text {arcsinh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )\right )}{128 a c^2} \]

input

Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^2,x]
 

output

((-1/128*I)*(7*Pi^4 + (8*I)*Pi^3*ArcSinh[a*x] + 24*Pi^2*ArcSinh[a*x]^2 + ( 
(192*I)*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2] - (32*I)*Pi*ArcSinh[a*x]^3 + ((6 
4*I)*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2) - 16*ArcSinh[a*x]^4 - 384*ArcSinh[a 
*x]*Log[1 - I/E^ArcSinh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcSinh[a*x]] + 384 
*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] + 48*Pi^2*ArcSinh[a*x]*Log[1 + I/E 
^ArcSinh[a*x]] - (96*I)*Pi*ArcSinh[a*x]^2*Log[1 + I/E^ArcSinh[a*x]] - 64*A 
rcSinh[a*x]^3*Log[1 + I/E^ArcSinh[a*x]] - 48*Pi^2*ArcSinh[a*x]*Log[1 - I*E 
^ArcSinh[a*x]] + (96*I)*Pi*ArcSinh[a*x]^2*Log[1 - I*E^ArcSinh[a*x]] - (8*I 
)*Pi^3*Log[1 + I*E^ArcSinh[a*x]] + 64*ArcSinh[a*x]^3*Log[1 + I*E^ArcSinh[a 
*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcSinh[a*x])/4]] - 48*(8 + Pi^2 - ( 
4*I)*Pi*ArcSinh[a*x] - 4*ArcSinh[a*x]^2)*PolyLog[2, (-I)/E^ArcSinh[a*x]] + 
 384*PolyLog[2, I/E^ArcSinh[a*x]] + 192*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^A 
rcSinh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*ArcSinh[a 
*x]*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcSinh[a* 
x]] + 384*ArcSinh[a*x]*PolyLog[3, (-I)/E^ArcSinh[a*x]] - 384*ArcSinh[a*x]* 
PolyLog[3, (-I)*E^ArcSinh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcSinh[a*x]] 
+ 384*PolyLog[4, (-I)/E^ArcSinh[a*x]] + 384*PolyLog[4, (-I)*E^ArcSinh[a*x] 
]))/(a*c^2)
 

3.4.32.3 Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.86, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {6203, 27, 6204, 3042, 4668, 3011, 6213, 6204, 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.

\(\displaystyle \int \frac {\text {arcsinh}(a x)^3}{\left (a^2 c x^2+c\right )^2} \, dx\)

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSinh[a*x]^3/(c + a^2*c*x^2)^2,x]
 

output

(x*ArcSinh[a*x]^3)/(2*c^2*(1 + a^2*x^2)) - (3*a*(-(ArcSinh[a*x]^2/(a^2*Sqr 
t[1 + a^2*x^2])) + (2*(2*ArcSinh[a*x]*ArcTan[E^ArcSinh[a*x]] - I*PolyLog[2 
, (-I)*E^ArcSinh[a*x]] + I*PolyLog[2, I*E^ArcSinh[a*x]]))/a^2))/(2*c^2) + 
(2*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]] + (3*I)*(-(ArcSinh[a*x]^2*PolyLog 
[2, (-I)*E^ArcSinh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x] 
] - PolyLog[4, (-I)*E^ArcSinh[a*x]])) - (3*I)*(-(ArcSinh[a*x]^2*PolyLog[2, 
 I*E^ArcSinh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]] - PolyL 
og[4, I*E^ArcSinh[a*x]])))/(2*a*c^2)
 

3.4.32.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 6203

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + 
c^2*x^2)^p]   Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

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]
 

3.4.32.4 Maple [F]

\[\int \frac {\operatorname {arcsinh}\left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2}}d x\]

input

int(arcsinh(a*x)^3/(a^2*c*x^2+c)^2,x)
 

output

int(arcsinh(a*x)^3/(a^2*c*x^2+c)^2,x)
 

3.4.32.5 Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

input

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^2,x, algorithm="fricas")
 

output

integral(arcsinh(a*x)^3/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x)
 

3.4.32.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

input

integrate(asinh(a*x)**3/(a**2*c*x**2+c)**2,x)
 

output

Integral(asinh(a*x)**3/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2
 

3.4.32.7 Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

input

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^2,x, algorithm="maxima")
 

output

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c)^2, x)
 

3.4.32.8 Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

input

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^2,x, algorithm="giac")
 

output

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c)^2, x)
 

3.4.32.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

input

int(asinh(a*x)^3/(c + a^2*c*x^2)^2,x)
 

output

int(asinh(a*x)^3/(c + a^2*c*x^2)^2, x)