\(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) [279]

Optimal result

Integrand size = 28, antiderivative size = 378 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {4 b^2 c^3 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \] Output:

-1/3*b^2*c^2*d*(c^2*d*x^2+d)^(1/2)/x+1/3*b^2*c^3*d*(c^2*d*x^2+d)^(1/2)*arc 
sinh(c*x)/(c^2*x^2+1)^(1/2)-1/3*b*c*d*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2 
)*(a+b*arcsinh(c*x))/x^2-c^2*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/x- 
4/3*c^3*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2)-1/3*( 
c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/x^3+1/3*c^3*d*(c^2*d*x^2+d)^(1/2)* 
(a+b*arcsinh(c*x))^3/b/(c^2*x^2+1)^(1/2)+8/3*b*c^3*d*(c^2*d*x^2+d)^(1/2)*( 
a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))^2)/(c^2*x^2+1)^(1/2)+4/3*b^ 
2*c^3*d*(c^2*d*x^2+d)^(1/2)*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)/(c^2*x^2+ 
1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {-a b c d x \sqrt {d+c^2 d x^2}-a^2 d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-4 a^2 c^2 d x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-b^2 c^2 d x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+b d \sqrt {d+c^2 d x^2} \left (3 a c^3 x^3-b \left (-4 c^3 x^3+\sqrt {1+c^2 x^2}+4 c^2 x^2 \sqrt {1+c^2 x^2}\right )\right ) \text {arcsinh}(c x)^2+b^2 c^3 d x^3 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3+b d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \left (-b c x-2 a \sqrt {1+c^2 x^2} \left (1+4 c^2 x^2\right )+8 b c^3 x^3 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )+8 a b c^3 d x^3 \sqrt {d+c^2 d x^2} \log (c x)+3 a^2 c^3 d^{3/2} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-4 b^2 c^3 d x^3 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 x^3 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(-(a*b*c*d*x*Sqrt[d + c^2*d*x^2]) - a^2*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d 
*x^2] - 4*a^2*c^2*d*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - b^2*c^2*d* 
x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + b*d*Sqrt[d + c^2*d*x^2]*(3*a*c 
^3*x^3 - b*(-4*c^3*x^3 + Sqrt[1 + c^2*x^2] + 4*c^2*x^2*Sqrt[1 + c^2*x^2])) 
*ArcSinh[c*x]^2 + b^2*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + b*d*S 
qrt[d + c^2*d*x^2]*ArcSinh[c*x]*(-(b*c*x) - 2*a*Sqrt[1 + c^2*x^2]*(1 + 4*c 
^2*x^2) + 8*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])]) + 8*a*b*c^3*d*x^3*Sqrt 
[d + c^2*d*x^2]*Log[c*x] + 3*a^2*c^3*d^(3/2)*x^3*Sqrt[1 + c^2*x^2]*Log[c*d 
*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 4*b^2*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*Po 
lyLog[2, E^(-2*ArcSinh[c*x])])/(3*x^3*Sqrt[1 + c^2*x^2])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.92 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.10, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6222, 6217, 247, 222, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838, 6220, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838, 6198}

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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 6217

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6220

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 222

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

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

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 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6198

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

rule 6217

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

rule 6220

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e* 
x^2]/Sqrt[1 + c^2*x^2]]   Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x] - Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] 
Int[(f*x)^(m + 2)*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x]) /; Fr 
eeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1]
 

rule 6222

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*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m 
 + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1928\) vs. \(2(350)=700\).

Time = 1.32 (sec) , antiderivative size = 1929, normalized size of antiderivative = 5.10

method	result	size
default	\(\text {Expression too large to display}\)	\(1929\)
parts	\(\text {Expression too large to display}\)	\(1929\)

Input:

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

Output:

-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/x^2/(c^2*x^2+1)^ 
(1/2)*arcsinh(x*c)*c-4/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2 
+1)*x^3*c^6-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/x^3/( 
c^2*x^2+1)*arcsinh(x*c)^2+8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)* 
arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))*d*c^3-20/3*b^2*(d*(c^2*x^2+1))^(1 
/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-29/3*b^2*(d*(c^2*x^2+1) 
)^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6+4/3*b^2*(d*(c^2*x^2 
+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x*arcsinh(x*c)*c^4+16/3*b^2*(d*(c^2* 
x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^3*arcsinh(x*c)*c^6+3*b^2*(d*(c^ 
2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+4/3*b 
^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcs 
inh(x*c)^2*c^3-3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9*c^2*x^2+1)/(c^2 
*x^2+1)^(1/2)*arcsinh(x*c)*c^3+8*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24*c^4*x^4+9 
*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7-10/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(24 
*c^4*x^4+9*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(2 
4*c^4*x^4+9*c^2*x^2+1)/x/(c^2*x^2+1)*c^2+8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^ 
2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*d*c^3+a^2*c^4*d*x* 
(c^2*d*x^2+d)^(1/2)+a^2*c^4*d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/ 
2))/(c^2*d)^(1/2)-2/3*a^2*c^2/d/x*(c^2*d*x^2+d)^(5/2)-1/3*a^2/d/x^3*(c^2*d 
*x^2+d)^(5/2)+2/3*a^2*c^4*x*(c^2*d*x^2+d)^(3/2)-4/3*b^2*(d*(c^2*x^2+1))...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*d*( - 4*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt(c**2*x**2 + 1)* 
a**2 + 6*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**4,x)*a*b*x**3 + 6*int((sq 
rt(c**2*x**2 + 1)*asinh(c*x))/x**2,x)*a*b*c**2*x**3 + 3*int((sqrt(c**2*x** 
2 + 1)*asinh(c*x)**2)/x**4,x)*b**2*x**3 + 3*int((sqrt(c**2*x**2 + 1)*asinh 
(c*x)**2)/x**2,x)*b**2*c**2*x**3 + 3*log(sqrt(c**2*x**2 + 1) + c*x)*a**2*c 
**3*x**3))/(3*x**3)