\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 (d+c^2 d x^2)^2} \, dx\) [252]

Optimal result

Integrand size = 26, antiderivative size = 287 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2} \] Output:

-b*c*(a+b*arcsinh(c*x))/d^2/(c^2*x^2+1)^(1/2)-(a+b*arcsinh(c*x))^2/d^2/x/( 
c^2*x^2+1)-3/2*c^2*x*(a+b*arcsinh(c*x))^2/d^2/(c^2*x^2+1)-3*c*(a+b*arcsinh 
(c*x))^2*arctan(c*x+(c^2*x^2+1)^(1/2))/d^2+b^2*c*arctan(c*x)/d^2-4*b*c*(a+ 
b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/d^2-2*b^2*c*polylog(2,-c*x- 
(c^2*x^2+1)^(1/2))/d^2+3*I*b*c*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x 
^2+1)^(1/2)))/d^2-3*I*b*c*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^ 
(1/2)))/d^2+2*b^2*c*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d^2-3*I*b^2*c*polylog 
(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^2+3*I*b^2*c*polylog(3,I*(c*x+(c^2*x^2+1)^ 
(1/2)))/d^2
 

Mathematica [A] (verified)

Time = 6.94 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {a^2}{d^2 x}-\frac {a^2 c^2 x}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 a^2 c \arctan (c x)}{2 d^2}+\frac {2 a b c \left (\frac {\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)}{4 (-1-i c x)}-\frac {\text {arcsinh}(c x)}{c x}-\frac {i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)}{4 (i+c x)}-\text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {3}{4} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {3}{4} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )\right )}{d^2}+\frac {b^2 c \left (-\frac {2 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {c x \text {arcsinh}(c x)^2}{1+c^2 x^2}+4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-\text {arcsinh}(c x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+3 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-3 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+6 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-6 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )+\text {arcsinh}(c x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{2 d^2} \] Input:

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

Output:

-(a^2/(d^2*x)) - (a^2*c^2*x)/(2*d^2*(1 + c^2*x^2)) - (3*a^2*c*ArcTan[c*x]) 
/(2*d^2) + (2*a*b*c*((Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x])/(4*(-1 - I*c*x)) 
 - ArcSinh[c*x]/(c*x) - (I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x])/(4*(I + c*x)) 
 - ArcTanh[Sqrt[1 + c^2*x^2]] + ((3*I)/4)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh 
[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 2*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((3 
*I)/4)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2 
*PolyLog[2, I*E^ArcSinh[c*x]])))/d^2 + (b^2*c*((-2*ArcSinh[c*x])/Sqrt[1 + 
c^2*x^2] - (c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) + 4*ArcTan[Tanh[ArcSinh[c*x] 
/2]] - ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] + 4*ArcSinh[c*x]*Log[1 - E^(-Ar 
cSinh[c*x])] + (3*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - (3*I)*ArcS 
inh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[ 
c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] + (6*I)*ArcSinh[c*x]*PolyLog[2, 
(-I)/E^ArcSinh[c*x]] - (6*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - 4 
*PolyLog[2, E^(-ArcSinh[c*x])] + (6*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - ( 
6*I)*PolyLog[3, I/E^ArcSinh[c*x]] + ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/ 
(2*d^2)
 

Rubi [A] (verified)

Time = 3.30 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.06, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {6224, 27, 6203, 6204, 3042, 4668, 3011, 2720, 6213, 216, 6226, 216, 6231, 3042, 26, 4670, 2715, 2838, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6203

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((a + b*ArcSinh[c*x])^2/(d^2*x*(1 + c^2*x^2))) + (2*b*c*((a + b*ArcSinh[c 
*x])/Sqrt[1 + c^2*x^2] - b*ArcTan[c*x] + I*((2*I)*(a + b*ArcSinh[c*x])*Arc 
Tanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ 
ArcSinh[c*x]])))/d^2 - (3*c^2*((x*(a + b*ArcSinh[c*x])^2)/(2*(1 + c^2*x^2) 
) - b*c*(-((a + b*ArcSinh[c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[c*x]) 
/c^2) + (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-((a + 
 b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + b*PolyLog[3, (-I)*E^Ar 
cSinh[c*x]]) - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x] 
]) + b*PolyLog[3, I*E^ArcSinh[c*x]]))/(2*c)))/d^2
 

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 27

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

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 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 6224

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

rule 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a**2/(c**4*x**6 + 2*c**2*x**4 + x**2), x) + Integral(b**2*asinh( 
c*x)**2/(c**4*x**6 + 2*c**2*x**4 + x**2), x) + Integral(2*a*b*asinh(c*x)/( 
c**4*x**6 + 2*c**2*x**4 + x**2), x))/d**2
 

Maxima [F]

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

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

Output:

-1/2*a^2*((3*c^2*x^2 + 2)/(c^2*d^2*x^3 + d^2*x) + 3*c*arctan(c*x)/d^2) + i 
ntegrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + 
 d^2*x^2) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^6 + 2*c^2*d^2*x^ 
4 + d^2*x^2), x)
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*atan(c*x)*a**2*c**3*x**3 - 3*atan(c*x)*a**2*c*x + 4*int(asinh(c*x)/( 
c**4*x**6 + 2*c**2*x**4 + x**2),x)*a*b*c**2*x**3 + 4*int(asinh(c*x)/(c**4* 
x**6 + 2*c**2*x**4 + x**2),x)*a*b*x + 2*int(asinh(c*x)**2/(c**4*x**6 + 2*c 
**2*x**4 + x**2),x)*b**2*c**2*x**3 + 2*int(asinh(c*x)**2/(c**4*x**6 + 2*c* 
*2*x**4 + x**2),x)*b**2*x - 3*a**2*c**2*x**2 - 2*a**2)/(2*d**2*x*(c**2*x** 
2 + 1))