\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x (d+c^2 d x^2)^{5/2}} \, dx\) [323]

Optimal result

Integrand size = 28, antiderivative size = 518 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c x (a+b \text {arcsinh}(c x))}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {(a+b \text {arcsinh}(c x))^2}{d^2 \sqrt {d+c^2 d x^2}}-\frac {14 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {7 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {7 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \] Output:

-1/3*b^2/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*c*x*(a+b*arcsinh(c*x))/d^2/(c^2*x^2 
+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+1/3*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(3/ 
2)+(a+b*arcsinh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)-14/3*b*(c^2*x^2+1)^(1/2)*( 
a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)-2* 
(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2*arctanh(c*x+(c^2*x^2+1)^(1/2))/d^2/ 
(c^2*d*x^2+d)^(1/2)-2*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,-c* 
x-(c^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)+7/3*I*b^2*(c^2*x^2+1)^(1/2)*p 
olylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^2/(c^2*d*x^2+d)^(1/2)-7/3*I*b^2*(c^ 
2*x^2+1)^(1/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d^2/(c^2*d*x^2+d)^(1/2 
)+2*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,c*x+(c^2*x^2+1)^(1/2) 
)/d^2/(c^2*d*x^2+d)^(1/2)+2*b^2*(c^2*x^2+1)^(1/2)*polylog(3,-c*x-(c^2*x^2+ 
1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)-2*b^2*(c^2*x^2+1)^(1/2)*polylog(3,c*x+(c 
^2*x^2+1)^(1/2))/d^2/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 2.94 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 \left (4+3 c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{\left (1+c^2 x^2\right )^2}+3 a^2 \sqrt {d} \log (c x)-3 a^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {a b d^2 \left (1+c^2 x^2\right )^{3/2} \left (-\frac {c x}{1+c^2 x^2}+\frac {2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {6 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-14 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+6 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-6 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+6 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-6 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\left (d+c^2 d x^2\right )^{3/2}}+\frac {b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \left (-\frac {1}{\sqrt {1+c^2 x^2}}-\frac {c x \text {arcsinh}(c x)}{1+c^2 x^2}+\frac {\text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^{3/2}}+\frac {3 \text {arcsinh}(c x)^2}{\sqrt {1+c^2 x^2}}+3 \text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )+7 i \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-7 i \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-3 \text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+7 i \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-7 i \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-6 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )\right )}{\left (d+c^2 d x^2\right )^{3/2}}}{3 d^3} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^(5/2)),x]
 

Output:

((a^2*(4 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2])/(1 + c^2*x^2)^2 + 3*a^2*Sqrt[d] 
*Log[c*x] - 3*a^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (a*b*d^2* 
(1 + c^2*x^2)^(3/2)*(-((c*x)/(1 + c^2*x^2)) + (2*ArcSinh[c*x])/(1 + c^2*x^ 
2)^(3/2) + (6*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 14*ArcTan[Tanh[ArcSinh[c*x 
]/2]] + 6*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 6*ArcSinh[c*x]*Log[1 + 
 E^(-ArcSinh[c*x])] + 6*PolyLog[2, -E^(-ArcSinh[c*x])] - 6*PolyLog[2, E^(- 
ArcSinh[c*x])]))/(d + c^2*d*x^2)^(3/2) + (b^2*d^2*(1 + c^2*x^2)^(3/2)*(-(1 
/Sqrt[1 + c^2*x^2]) - (c*x*ArcSinh[c*x])/(1 + c^2*x^2) + ArcSinh[c*x]^2/(1 
 + c^2*x^2)^(3/2) + (3*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]^ 
2*Log[1 - E^(-ArcSinh[c*x])] + (7*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x] 
] - (7*I)*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - 3*ArcSinh[c*x]^2*Log[1 
+ E^(-ArcSinh[c*x])] + 6*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] + (7* 
I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (7*I)*PolyLog[2, I/E^ArcSinh[c*x]] - 
6*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 6*PolyLog[3, -E^(-ArcSinh[c 
*x])] - 6*PolyLog[3, E^(-ArcSinh[c*x])]))/(d + c^2*d*x^2)^(3/2))/(3*d^3)
 

Rubi [A] (verified)

Time = 4.25 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.81, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6226, 6203, 241, 6204, 3042, 4668, 2715, 2838, 6226, 6204, 3042, 4668, 2715, 2838, 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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^(5/2)),x]
 

Output:

(a + b*ArcSinh[c*x])^2/(3*d*(d + c^2*d*x^2)^(3/2)) - (2*b*c*Sqrt[1 + c^2*x 
^2]*(b/(2*c*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(2*(1 + c^2*x^2) 
) + (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E 
^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c)))/(3*d^2*Sqrt[d + 
 c^2*d*x^2]) + ((a + b*ArcSinh[c*x])^2/(d*Sqrt[d + c^2*d*x^2]) - (2*b*Sqrt 
[1 + c^2*x^2]*(2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog 
[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]]))/(d*Sqrt[d + 
c^2*d*x^2]) + (I*Sqrt[1 + c^2*x^2]*((2*I)*(a + b*ArcSinh[c*x])^2*ArcTanh[E 
^ArcSinh[c*x]] - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x 
]]) + b*PolyLog[3, -E^ArcSinh[c*x]]) + (2*I)*b*(-((a + b*ArcSinh[c*x])*Pol 
yLog[2, E^ArcSinh[c*x]]) + b*PolyLog[3, E^ArcSinh[c*x]])))/(d*Sqrt[d + c^2 
*d*x^2]))/d
 

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 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -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 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 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 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 6226

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ 
b*c*(n/(2*f*(p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

Input:

int((a+b*arcsinh(x*c))^2/x/(c^2*d*x^2+d)^(5/2),x)
 

Output:

int((a+b*arcsinh(x*c))^2/x/(c^2*d*x^2+d)^(5/2),x)
 

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas" 
)
 

Output:

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^ 
2)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*asinh(c*x))**2/x/(c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*asinh(c*x))**2/(x*(d*(c**2*x**2 + 1))**(5/2)), x)
 

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima" 
)
 

Output:

-1/3*a^2*(3*arcsinh(1/(c*abs(x)))/d^(5/2) - 3/(sqrt(c^2*d*x^2 + d)*d^2) - 
1/((c^2*d*x^2 + d)^(3/2)*d)) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^ 
2/((c^2*d*x^2 + d)^(5/2)*x) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/((c^2*d*x 
^2 + d)^(5/2)*x), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^(5/2)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((a + b*asinh(c*x))^2/(x*(d + c^2*d*x^2)^(5/2)),x)
 

Output:

int((a + b*asinh(c*x))^2/(x*(d + c^2*d*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+4 \sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{5}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {c^{2} x^{2}+1}\, x}d x \right ) a b \,c^{4} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{5}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {c^{2} x^{2}+1}\, x}d x \right ) a b \,c^{2} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{5}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {c^{2} x^{2}+1}\, x}d x \right ) a b +3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{5}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {c^{2} x^{2}+1}\, x}d x \right ) b^{2} c^{4} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{5}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {c^{2} x^{2}+1}\, x}d x \right ) b^{2} c^{2} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{5}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {c^{2} x^{2}+1}\, x}d x \right ) b^{2}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2} c^{4} x^{4}+6 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2} c^{4} x^{4}-6 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2} c^{2} x^{2}-3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2}}{3 \sqrt {d}\, d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))^2/x/(c^2*d*x^2+d)^(5/2),x)
 

Output:

(3*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 + 4*sqrt(c**2*x**2 + 1)*a**2 + 6*int 
(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**5 + 2*sqrt(c**2*x**2 + 1)*c**2*x* 
*3 + sqrt(c**2*x**2 + 1)*x),x)*a*b*c**4*x**4 + 12*int(asinh(c*x)/(sqrt(c** 
2*x**2 + 1)*c**4*x**5 + 2*sqrt(c**2*x**2 + 1)*c**2*x**3 + sqrt(c**2*x**2 + 
 1)*x),x)*a*b*c**2*x**2 + 6*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**5 
+ 2*sqrt(c**2*x**2 + 1)*c**2*x**3 + sqrt(c**2*x**2 + 1)*x),x)*a*b + 3*int( 
asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**4*x**5 + 2*sqrt(c**2*x**2 + 1)*c**2* 
x**3 + sqrt(c**2*x**2 + 1)*x),x)*b**2*c**4*x**4 + 6*int(asinh(c*x)**2/(sqr 
t(c**2*x**2 + 1)*c**4*x**5 + 2*sqrt(c**2*x**2 + 1)*c**2*x**3 + sqrt(c**2*x 
**2 + 1)*x),x)*b**2*c**2*x**2 + 3*int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c 
**4*x**5 + 2*sqrt(c**2*x**2 + 1)*c**2*x**3 + sqrt(c**2*x**2 + 1)*x),x)*b** 
2 + 3*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a**2*c**4*x**4 + 6*log(sqrt(c**2* 
x**2 + 1) + c*x - 1)*a**2*c**2*x**2 + 3*log(sqrt(c**2*x**2 + 1) + c*x - 1) 
*a**2 - 3*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a**2*c**4*x**4 - 6*log(sqrt(c 
**2*x**2 + 1) + c*x + 1)*a**2*c**2*x**2 - 3*log(sqrt(c**2*x**2 + 1) + c*x 
+ 1)*a**2)/(3*sqrt(d)*d**2*(c**4*x**4 + 2*c**2*x**2 + 1))