\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 (d+c^2 d x^2)^{3/2}} \, dx\) [315]

Optimal result

Integrand size = 28, antiderivative size = 573 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x \sqrt {d+c^2 d x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {d+c^2 d x^2}}+\frac {4 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b c^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 7.30 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Sqrt[d*(1 + c^2*x^2)]*(-1/2*a^2/(d^2*x^2) - (a^2*c^2)/(d^2*(1 + c^2*x^2))) 
 - (3*a^2*c^2*Log[x])/(2*d^(3/2)) + (3*a^2*c^2*Log[d + Sqrt[d]*Sqrt[d*(1 + 
 c^2*x^2)]])/(2*d^(3/2)) + (a*b*c^2*(-8*ArcSinh[c*x] + 16*Sqrt[1 + c^2*x^2 
]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] 
- Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2* 
x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*ArcSin 
h[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 12*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-A 
rcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - Sqrt[ 
1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Sqrt[1 + c^2*x^2]*Tan 
h[ArcSinh[c*x]/2]))/(4*d*Sqrt[d*(1 + c^2*x^2)]) + (b^2*c^2*(-8*ArcSinh[c*x 
]^2 - 4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2 
*x^2]*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh 
[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] 
*Log[1 - I/E^ArcSinh[c*x]] + (16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + 
 I/E^ArcSinh[c*x]] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 + E^(-ArcSi 
nh[c*x])] + 8*Sqrt[1 + c^2*x^2]*Log[Tanh[ArcSinh[c*x]/2]] - 24*Sqrt[1 + c^ 
2*x^2]*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x 
^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (16*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, 
I/E^ArcSinh[c*x]] + 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, E^(-ArcSi 
nh[c*x])] - 24*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^(-ArcSinh[c*x])] + 24*Sq...
 

Rubi [A] (verified)

Time = 5.02 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.71, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {6224, 6224, 243, 73, 221, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {3}{2} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^{3/2}}dx+\frac {b c \sqrt {c^2 x^2+1} \left (c^2 \left (-\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx\right )+\frac {b \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c}-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2 \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*(a + b*ArcSinh[c*x])^2/(d*x^2*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[1 + c^ 
2*x^2]*(-((a + b*ArcSinh[c*x])/x) - b*c*ArcTanh[Sqrt[1 + c^2*x^2]] - c*(2* 
(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSin 
h[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])))/(d*Sqrt[d + c^2*d*x^2]) - (3 
*c^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*Po 
lyLog[3, -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]])))/(d*Sqrt[d + c^2*d*x^2]))) 
/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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 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 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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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