\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 (d+c^2 d x^2)^{3/2}} \, dx\) [316]

Optimal result

Integrand size = 28, antiderivative size = 443 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 d^2 x}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {16 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}} \] Output:

-1/3*b^2*c^2*(c^2*d*x^2+d)^(1/2)/d^2/x-1/3*b*c*(c^2*x^2+1)^(1/2)*(a+b*arcs 
inh(c*x))/d/x^2/(c^2*d*x^2+d)^(1/2)-1/3*(a+b*arcsinh(c*x))^2/d/x^3/(c^2*d* 
x^2+d)^(1/2)+4/3*c^2*(a+b*arcsinh(c*x))^2/d/x/(c^2*d*x^2+d)^(1/2)+8/3*c^4* 
x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(1/2)+8/3*c^3*(c^2*x^2+1)^(1/2)*(a+ 
b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(1/2)+20/3*b*c^3*(c^2*x^2+1)^(1/2)*(a+b* 
arcsinh(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d/(c^2*d*x^2+d)^(1/2)-16/ 
3*b*c^3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^ 
2)/d/(c^2*d*x^2+d)^(1/2)-b^2*c^3*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^ 
2+1)^(1/2))^2)/d/(c^2*d*x^2+d)^(1/2)-5/3*b^2*c^3*(c^2*x^2+1)^(1/2)*polylog 
(2,(c*x+(c^2*x^2+1)^(1/2))^2)/d/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-a^2+4 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4-b^2 c^4 x^4-a b c x \sqrt {1+c^2 x^2}-2 a b \text {arcsinh}(c x)+8 a b c^2 x^2 \text {arcsinh}(c x)+16 a b c^4 x^4 \text {arcsinh}(c x)-b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-b^2 \text {arcsinh}(c x)^2+4 b^2 c^2 x^2 \text {arcsinh}(c x)^2+8 b^2 c^4 x^4 \text {arcsinh}(c x)^2-8 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-10 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-6 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-10 a b c^3 x^3 \sqrt {1+c^2 x^2} \log (c x)-3 a b c^3 x^3 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+3 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+5 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 d x^3 \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(3/2)),x]
 

Output:

(-a^2 + 4*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 - b^2*c^4*x^4 - a*b*c* 
x*Sqrt[1 + c^2*x^2] - 2*a*b*ArcSinh[c*x] + 8*a*b*c^2*x^2*ArcSinh[c*x] + 16 
*a*b*c^4*x^4*ArcSinh[c*x] - b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b^2*A 
rcSinh[c*x]^2 + 4*b^2*c^2*x^2*ArcSinh[c*x]^2 + 8*b^2*c^4*x^4*ArcSinh[c*x]^ 
2 - 8*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - 10*b^2*c^3*x^3*Sqrt[1 
 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 6*b^2*c^3*x^3*Sqrt 
[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 10*a*b*c^3*x^3*S 
qrt[1 + c^2*x^2]*Log[c*x] - 3*a*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^ 
2] + 3*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 5* 
b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*d*x^3*Sq 
rt[d + c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.47 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.01, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6224, 6224, 242, 6202, 6212, 3042, 26, 4201, 2620, 2715, 2838, 6214, 5984, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 6202

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {4}{3} c^2 \left (\frac {4 i b c \sqrt {c^2 x^2+1} \left (\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d \sqrt {c^2 d x^2+d}}-2 c^2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \sqrt {c^2 d x^2+d}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 i c^2 \left (\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2}-\frac {b c \sqrt {c^2 x^2+1}}{2 x}\right )}{3 d \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(3/2)),x]
 

Output:

-1/3*(a + b*ArcSinh[c*x])^2/(d*x^3*Sqrt[d + c^2*d*x^2]) + (2*b*c*Sqrt[1 + 
c^2*x^2]*(-1/2*(b*c*Sqrt[1 + c^2*x^2])/x - (a + b*ArcSinh[c*x])/(2*x^2) - 
(2*I)*c^2*(I*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*Po 
lyLog[2, -E^(2*ArcSinh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x])])))/ 
(3*d*Sqrt[d + c^2*d*x^2]) - (4*c^2*(-((a + b*ArcSinh[c*x])^2/(d*x*Sqrt[d + 
 c^2*d*x^2])) - 2*c^2*((x*(a + b*ArcSinh[c*x])^2)/(d*Sqrt[d + c^2*d*x^2]) 
+ ((2*I)*b*Sqrt[1 + c^2*x^2]*(((-1/2*I)*(a + b*ArcSinh[c*x])^2)/b + (2*I)* 
(((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/2 + (b*PolyLog[2, -E^( 
2*ArcSinh[c*x])])/4)))/(c*d*Sqrt[d + c^2*d*x^2])) + ((4*I)*b*c*Sqrt[1 + c^ 
2*x^2]*(I*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyL 
og[2, -E^(2*ArcSinh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x])]))/(d*S 
qrt[d + c^2*d*x^2])))/3
 

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 242

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] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && 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 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 6202

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

rule 6212

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

rule 6214

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[1/d   Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar 
cSinh[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2607\) vs. \(2(429)=858\).

Time = 1.30 (sec) , antiderivative size = 2608, normalized size of antiderivative = 5.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2608\)
parts	\(\text {Expression too large to display}\)	\(2608\)

Input:

int((a+b*arcsinh(x*c))^2/x^4/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x*c^2+1/3*b^2*(d 
*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x^3*arcsinh(x*c)^2+16/3*b^ 
2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*arcsinh(x*c)^2*c^3-b^2*(d*(c 
^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^ 
2)*c^3-10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2*polylog(2,-x*c 
-(c^2*x^2+1)^(1/2))*c^3-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2 
-1)/d^2*c^3*(c^2*x^2+1)^(1/2)-10/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^( 
1/2)/d^2*polylog(2,x*c+(c^2*x^2+1)^(1/2))*c^3+32/3*b^2*(d*(c^2*x^2+1))^(1/ 
2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^7*c^10+128/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8 
*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*arcsinh(x*c)*c^6+8/3*a*b*(d*(c^2*x^2+1))^(1/ 
2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x*(c^2*x^2+1)*c^4+16*a*b*(d*(c^2*x^2+1))^(1 
/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x*arcsinh(x*c)*c^4+16/3*a*b*(d*(c^2*x^2+1) 
)^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^3-8*a 
*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x*arcsinh(x*c)*c^2+1/ 
3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2/x^2*(c^2*x^2+1)^(1 
/2)*c-32/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^3*(c^2* 
x^2+1)*arcsinh(x*c)*c^6-64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^ 
2-1)/d^2*x^2*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*c^5+8/3*b^2*(d*(c^2*x^2+1))^ 
(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x*(c^2*x^2+1)*arcsinh(x*c)*c^4-64/3*b^2* 
(d*(c^2*x^2+1))^(1/2)/(8*c^4*x^4+7*c^2*x^2-1)/d^2*x^5*(c^2*x^2+1)*arcsi...
 

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:

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

Sympy [F]

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

integrate((a+b*asinh(c*x))**2/x**4/(c**2*d*x**2+d)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/x^4/(c^2*d*x^2+d)^(3/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:

int((a + b*asinh(c*x))^2/(x^4*(d + c^2*d*x^2)^(3/2)),x)
 

Output:

int((a + b*asinh(c*x))^2/(x^4*(d + c^2*d*x^2)^(3/2)), x)
 

Reduce [F]

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

int((a+b*asinh(c*x))^2/x^4/(c^2*d*x^2+d)^(3/2),x)
 

Output:

(8*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 + 4*sqrt(c**2*x**2 + 1)*a**2*c**2*x* 
*2 - sqrt(c**2*x**2 + 1)*a**2 + 6*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2 
*x**6 + sqrt(c**2*x**2 + 1)*x**4),x)*a*b*c**2*x**5 + 6*int(asinh(c*x)/(sqr 
t(c**2*x**2 + 1)*c**2*x**6 + sqrt(c**2*x**2 + 1)*x**4),x)*a*b*x**3 + 3*int 
(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**2*x**6 + sqrt(c**2*x**2 + 1)*x**4), 
x)*b**2*c**2*x**5 + 3*int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**2*x**6 + s 
qrt(c**2*x**2 + 1)*x**4),x)*b**2*x**3 - 8*a**2*c**5*x**5 - 8*a**2*c**3*x** 
3)/(3*sqrt(d)*d*x**3*(c**2*x**2 + 1))