\(\int \frac {(d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx\) [225]

Optimal result

Integrand size = 26, antiderivative size = 229 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {32}{9} b^2 c^2 d^2 x+\frac {2}{27} b^2 c^4 d^2 x^3-\frac {10}{3} b c d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {2}{9} b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {8}{3} c^2 d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{3} c^2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-4 b c d^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-2 b^2 c d^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+2 b^2 c d^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \] Output:

32/9*b^2*c^2*d^2*x+2/27*b^2*c^4*d^2*x^3-10/3*b*c*d^2*(c^2*x^2+1)^(1/2)*(a+ 
b*arcsinh(c*x))-2/9*b*c*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))+8/3*c^2*d 
^2*x*(a+b*arcsinh(c*x))^2+4/3*c^2*d^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2-d 
^2*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/x-4*b*c*d^2*(a+b*arcsinh(c*x))*arcta 
nh(c*x+(c^2*x^2+1)^(1/2))-2*b^2*c*d^2*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2* 
b^2*c*d^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {1}{54} d^2 \left (-\frac {54 a^2}{x}+108 a^2 c^2 x+18 a^2 c^4 x^3-12 a b c \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2}+36 a b c^4 x^3 \text {arcsinh}(c x)-189 b^2 c \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+216 a b c \left (-\sqrt {1+c^2 x^2}+c x \text {arcsinh}(c x)\right )+108 b^2 c^2 x \left (2+\text {arcsinh}(c x)^2\right )+2 b^2 c^2 x \left (-12+2 c^2 x^2+9 c^2 x^2 \text {arcsinh}(c x)^2\right )-\frac {108 a b \left (\text {arcsinh}(c x)+c x \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )\right )}{x}-3 b^2 c \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x))-\frac {54 b^2 \text {arcsinh}(c x) \left (\text {arcsinh}(c x)+2 c x \left (-\log \left (1-e^{-\text {arcsinh}(c x)}\right )+\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )\right )}{x}+108 b^2 c \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-108 b^2 c \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right ) \] Input:

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

Output:

(d^2*((-54*a^2)/x + 108*a^2*c^2*x + 18*a^2*c^4*x^3 - 12*a*b*c*(-2 + c^2*x^ 
2)*Sqrt[1 + c^2*x^2] + 36*a*b*c^4*x^3*ArcSinh[c*x] - 189*b^2*c*Sqrt[1 + c^ 
2*x^2]*ArcSinh[c*x] + 216*a*b*c*(-Sqrt[1 + c^2*x^2] + c*x*ArcSinh[c*x]) + 
108*b^2*c^2*x*(2 + ArcSinh[c*x]^2) + 2*b^2*c^2*x*(-12 + 2*c^2*x^2 + 9*c^2* 
x^2*ArcSinh[c*x]^2) - (108*a*b*(ArcSinh[c*x] + c*x*ArcTanh[Sqrt[1 + c^2*x^ 
2]]))/x - 3*b^2*c*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] - (54*b^2*ArcSinh[c*x] 
*(ArcSinh[c*x] + 2*c*x*(-Log[1 - E^(-ArcSinh[c*x])] + Log[1 + E^(-ArcSinh[ 
c*x])])))/x + 108*b^2*c*PolyLog[2, -E^(-ArcSinh[c*x])] - 108*b^2*c*PolyLog 
[2, E^(-ArcSinh[c*x])]))/54
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.19 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.32, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {6222, 27, 6201, 6187, 6213, 24, 2009, 6223, 2009, 6221, 24, 6231, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 27

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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 6187

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

rule 6201

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

rule 6213

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

rule 6221

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 + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 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]
 

rule 6223

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

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(c \left (d^{2} a^{2} \left (\frac {x^{3} c^{3}}{3}+2 x c -\frac {1}{x c}\right )-\frac {2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}}{9}+\frac {32 b^{2} d^{2} x c}{9}+2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\frac {2 b^{2} d^{2} x^{3} c^{3}}{27}+2 b^{2} d^{2} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{x c}+\frac {b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{3}}{3}+2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2} x c -\frac {32 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{9}+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+2 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(364\)
default	\(c \left (d^{2} a^{2} \left (\frac {x^{3} c^{3}}{3}+2 x c -\frac {1}{x c}\right )-\frac {2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}}{9}+\frac {32 b^{2} d^{2} x c}{9}+2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\frac {2 b^{2} d^{2} x^{3} c^{3}}{27}+2 b^{2} d^{2} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\frac {b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{x c}+\frac {b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{3}}{3}+2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2} x c -\frac {32 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{9}+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+2 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(364\)
parts	\(d^{2} a^{2} \left (\frac {c^{4} x^{3}}{3}+2 c^{2} x -\frac {1}{x}\right )+\frac {32 b^{2} c^{2} d^{2} x}{9}-\frac {2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{3}}{9}+2 b^{2} c \,d^{2} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} c \,d^{2} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+\frac {2 b^{2} c^{4} d^{2} x^{3}}{27}+2 b^{2} d^{2} c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d^{2} c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\frac {32 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, c}{9}-\frac {b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{x}+\frac {b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{4}}{3}+2 b^{2} d^{2} \operatorname {arcsinh}\left (x c \right )^{2} x \,c^{2}+2 d^{2} a b c \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+2 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {c^{2} x^{2}+1}}{9}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\)	\(368\)

Input:

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

Output:

c*(d^2*a^2*(1/3*x^3*c^3+2*x*c-1/x/c)-2/9*b^2*d^2*arcsinh(x*c)*(c^2*x^2+1)^ 
(1/2)*x^2*c^2+32/9*b^2*d^2*x*c+2*b^2*d^2*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1) 
^(1/2))+2/27*b^2*d^2*x^3*c^3+2*b^2*d^2*polylog(2,x*c+(c^2*x^2+1)^(1/2))-2* 
b^2*d^2*polylog(2,-x*c-(c^2*x^2+1)^(1/2))-2*b^2*d^2*arcsinh(x*c)*ln(1+x*c+ 
(c^2*x^2+1)^(1/2))-b^2*d^2*arcsinh(x*c)^2/x/c+1/3*b^2*d^2*arcsinh(x*c)^2*x 
^3*c^3+2*b^2*d^2*arcsinh(x*c)^2*x*c-32/9*b^2*d^2*arcsinh(x*c)*(c^2*x^2+1)^ 
(1/2)+2*d^2*a*b*(1/3*arcsinh(x*c)*x^3*c^3+2*x*c*arcsinh(x*c)-arcsinh(x*c)/ 
x/c-1/9*x^2*c^2*(c^2*x^2+1)^(1/2)-16/9*(c^2*x^2+1)^(1/2)-arctanh(1/(c^2*x^ 
2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*a^2*c^4*d^2*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c 
^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c^4*d^2 + 2*b^2*c^2*d^2*x*arcsinh(c*x)^ 
2 + 4*b^2*c^2*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + 2*a^2*c^2*d^2*x 
 + 4*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c*d^2 - 2*(c*arcsinh(1/(c* 
abs(x))) + arcsinh(c*x)/x)*a*b*d^2 - a^2*d^2/x + 1/3*(b^2*c^4*d^2*x^4 - 3* 
b^2*d^2)*log(c*x + sqrt(c^2*x^2 + 1))^2/x - integrate(2/3*(b^2*c^7*d^2*x^6 
 + b^2*c^5*d^2*x^4 - 3*b^2*c^3*d^2*x^2 - 3*b^2*c*d^2 + (b^2*c^6*d^2*x^5 - 
3*b^2*c^2*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c^3*x^4 
+ c*x^2 + (c^2*x^3 + x)*sqrt(c^2*x^2 + 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {d^{2} \left (18 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-36 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c x +6 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}+36 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-18 \mathit {asinh} \left (c x \right ) a b -2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-32 \sqrt {c^{2} x^{2}+1}\, a b c x +9 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} x +9 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{4} x +18 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a b c x -18 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a b c x +3 a^{2} c^{4} x^{4}+18 a^{2} c^{2} x^{2}-9 a^{2}+36 b^{2} c^{2} x^{2}\right )}{9 x} \] Input:

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

Output:

(d**2*(18*asinh(c*x)**2*b**2*c**2*x**2 - 36*sqrt(c**2*x**2 + 1)*asinh(c*x) 
*b**2*c*x + 6*asinh(c*x)*a*b*c**4*x**4 + 36*asinh(c*x)*a*b*c**2*x**2 - 18* 
asinh(c*x)*a*b - 2*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 - 32*sqrt(c**2*x**2 + 
 1)*a*b*c*x + 9*int(asinh(c*x)**2/x**2,x)*b**2*x + 9*int(asinh(c*x)**2*x** 
2,x)*b**2*c**4*x + 18*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a*b*c*x - 18*log( 
sqrt(c**2*x**2 + 1) + c*x + 1)*a*b*c*x + 3*a**2*c**4*x**4 + 18*a**2*c**2*x 
**2 - 9*a**2 + 36*b**2*c**2*x**2))/(9*x)