\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 (d+c^2 d x^2)^2} \, dx\) [254]

Optimal result

Integrand size = 26, antiderivative size = 401 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b^2 c^2}{3 d^2 x}+\frac {2 b c^3 (a+b \text {arcsinh}(c x))}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c (a+b \text {arcsinh}(c x))}{3 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))^2}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {b^2 c^3 \arctan (c x)}{d^2}+\frac {26 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d^2}+\frac {13 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d^2}-\frac {5 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {5 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {13 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d^2}+\frac {5 i b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {5 i b^2 c^3 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2} \] Output:

-1/3*b^2*c^2/d^2/x+2/3*b*c^3*(a+b*arcsinh(c*x))/d^2/(c^2*x^2+1)^(1/2)-1/3* 
b*c*(a+b*arcsinh(c*x))/d^2/x^2/(c^2*x^2+1)^(1/2)-1/3*(a+b*arcsinh(c*x))^2/ 
d^2/x^3/(c^2*x^2+1)+5/3*c^2*(a+b*arcsinh(c*x))^2/d^2/x/(c^2*x^2+1)+5/2*c^4 
*x*(a+b*arcsinh(c*x))^2/d^2/(c^2*x^2+1)+5*c^3*(a+b*arcsinh(c*x))^2*arctan( 
c*x+(c^2*x^2+1)^(1/2))/d^2-b^2*c^3*arctan(c*x)/d^2+26/3*b*c^3*(a+b*arcsinh 
(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/d^2+13/3*b^2*c^3*polylog(2,-c*x-(c^2 
*x^2+1)^(1/2))/d^2-5*I*b*c^3*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x^2 
+1)^(1/2)))/d^2+5*I*b*c^3*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^ 
(1/2)))/d^2-13/3*b^2*c^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d^2+5*I*b^2*c^3* 
polylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^2-5*I*b^2*c^3*polylog(3,I*(c*x+(c^ 
2*x^2+1)^(1/2)))/d^2
 

Mathematica [A] (verified)

Time = 7.85 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^2),x]
 

Output:

-1/3*a^2/(d^2*x^3) + (2*a^2*c^2)/(d^2*x) + (a^2*c^4*x)/(2*d^2*(1 + c^2*x^2 
)) + (5*a^2*c^3*ArcTan[c*x])/(2*d^2) + (2*a*b*(-1/6*(c*Sqrt[1 + c^2*x^2])/ 
x^2 - (c^3*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(4*(-1 - I*c*x)) - ArcSin 
h[c*x]/(3*x^3) + (c^4*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(4*(I*c + c^2* 
x)) + (c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/6 - 2*c^2*(-(ArcSinh[c*x]/x) - c*Ar 
cTanh[Sqrt[1 + c^2*x^2]]) - ((5*I)/4)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcS 
inh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]] 
)/c) + ((5*I)/4)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ 
ArcSinh[c*x]])/c + (2*PolyLog[2, I*E^ArcSinh[c*x]])/c)))/d^2 + (b^2*c^3*(( 
24*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (12*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) 
 - 48*ArcTan[Tanh[ArcSinh[c*x]/2]] - 4*Coth[ArcSinh[c*x]/2] + 26*ArcSinh[c 
*x]^2*Coth[ArcSinh[c*x]/2] - 2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - (c*x* 
ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 104*ArcSinh[c*x]*Log[1 - E^(-Ar 
cSinh[c*x])] - (60*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (60*I)*Ar 
cSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 104*ArcSinh[c*x]*Log[1 + E^(-ArcS 
inh[c*x])] - 104*PolyLog[2, -E^(-ArcSinh[c*x])] - (120*I)*ArcSinh[c*x]*Pol 
yLog[2, (-I)/E^ArcSinh[c*x]] + (120*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh 
[c*x]] + 104*PolyLog[2, E^(-ArcSinh[c*x])] - (120*I)*PolyLog[3, (-I)/E^Arc 
Sinh[c*x]] + (120*I)*PolyLog[3, I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sech[Ar 
cSinh[c*x]/2]^2 - (8*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2]^4)/(c^3*x^3) +...
 

Rubi [A] (verified)

Time = 5.73 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.21, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {6224, 27, 6224, 264, 216, 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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (c^2 d x^2+d\right )^2} \, dx\)

\(\Big \downarrow \) 6224

\(\displaystyle -\frac {5}{3} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{d^2 x^2 \left (c^2 x^2+1\right )^2}dx+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 x^2+1\right )^{3/2}}dx}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 x^2+1\right )^{3/2}}dx}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6224

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx+\frac {1}{2} b c \left (c^2 \left (-\int \frac {1}{c^2 x^2+1}dx\right )-\frac {1}{x}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {5 c^2 \left (-3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6203

\(\displaystyle \frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {5 c^2 \left (-3 c^2 \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6204

\(\displaystyle \frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {5 c^2 \left (-3 c^2 \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {5 c^2 \left (2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-3 c^2 \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {5 c^2 \left (-3 c^2 \left (\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6213

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \int \frac {1}{c^2 x^2+1}dx}{c}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6226

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int \frac {1}{c^2 x^2+1}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int \frac {1}{c^2 x^2+1}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 6231

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {5 c^2 \left (2 b c \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {5 c^2 \left (2 b c \left (i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {5 c^2 \left (2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-3 c^2 \left (-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}+\frac {2 b c \left (-\frac {3}{2} c^2 \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (-c \arctan (c x)-\frac {1}{x}\right )\right )}{3 d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^2 x^3 \left (c^2 x^2+1\right )}\)

Input:

Int[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^2),x]
 

Output:

-1/3*(a + b*ArcSinh[c*x])^2/(d^2*x^3*(1 + c^2*x^2)) + (2*b*c*(-1/2*(a + b* 
ArcSinh[c*x])/(x^2*Sqrt[1 + c^2*x^2]) + (b*c*(-x^(-1) - c*ArcTan[c*x]))/2 
- (3*c^2*((a + b*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - b*ArcTan[c*x] + I*((2*I 
)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh 
[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]])))/2))/(3*d^2) - (5*c^2*(-((a + b* 
ArcSinh[c*x])^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])*ArcTanh[E^ArcSinh 
[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]) 
) - 3*c^2*((x*(a + b*ArcSinh[c*x])^2)/(2*(1 + c^2*x^2)) - b*c*(-((a + b*Ar 
cSinh[c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[c*x])/c^2) + (2*(a + b*Ar 
cSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-((a + b*ArcSinh[c*x])*Pol 
yLog[2, (-I)*E^ArcSinh[c*x]]) + b*PolyLog[3, (-I)*E^ArcSinh[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))))/(3*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 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, 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 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]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{x^{4} \left (c^{2} d \,x^{2}+d \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arcsinh(c*x))^2/x^4/(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^8 + 2* 
c^2*d^2*x^6 + d^2*x^4), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a**2/(c**4*x**8 + 2*c**2*x**6 + x**4), x) + Integral(b**2*asinh( 
c*x)**2/(c**4*x**8 + 2*c**2*x**6 + x**4), x) + Integral(2*a*b*asinh(c*x)/( 
c**4*x**8 + 2*c**2*x**6 + x**4), x))/d**2
 

Maxima [F]

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

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/6*(15*c^3*arctan(c*x)/d^2 + (15*c^4*x^4 + 10*c^2*x^2 - 2)/(c^2*d^2*x^5 + 
 d^2*x^3))*a^2 + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^4*d^2*x^8 
 + 2*c^2*d^2*x^6 + d^2*x^4) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2* 
x^8 + 2*c^2*d^2*x^6 + d^2*x^4), x)
 

Giac [F]

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

integrate((a+b*arcsinh(c*x))^2/x^4/(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^4), x)
 

Mupad [F(-1)]

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

int((a + b*asinh(c*x))^2/(x^4*(d + c^2*d*x^2)^2),x)
 

Output:

int((a + b*asinh(c*x))^2/(x^4*(d + c^2*d*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {15 \mathit {atan} \left (c x \right ) a^{2} c^{5} x^{5}+15 \mathit {atan} \left (c x \right ) a^{2} c^{3} x^{3}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{4} x^{8}+2 c^{2} x^{6}+x^{4}}d x \right ) a b \,c^{2} x^{5}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{4} x^{8}+2 c^{2} x^{6}+x^{4}}d x \right ) a b \,x^{3}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{4} x^{8}+2 c^{2} x^{6}+x^{4}}d x \right ) b^{2} c^{2} x^{5}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{4} x^{8}+2 c^{2} x^{6}+x^{4}}d x \right ) b^{2} x^{3}+15 a^{2} c^{4} x^{4}+10 a^{2} c^{2} x^{2}-2 a^{2}}{6 d^{2} x^{3} \left (c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))^2/x^4/(c^2*d*x^2+d)^2,x)
 

Output:

(15*atan(c*x)*a**2*c**5*x**5 + 15*atan(c*x)*a**2*c**3*x**3 + 12*int(asinh( 
c*x)/(c**4*x**8 + 2*c**2*x**6 + x**4),x)*a*b*c**2*x**5 + 12*int(asinh(c*x) 
/(c**4*x**8 + 2*c**2*x**6 + x**4),x)*a*b*x**3 + 6*int(asinh(c*x)**2/(c**4* 
x**8 + 2*c**2*x**6 + x**4),x)*b**2*c**2*x**5 + 6*int(asinh(c*x)**2/(c**4*x 
**8 + 2*c**2*x**6 + x**4),x)*b**2*x**3 + 15*a**2*c**4*x**4 + 10*a**2*c**2* 
x**2 - 2*a**2)/(6*d**2*x**3*(c**2*x**2 + 1))