\(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx\) [276]

Optimal result

Integrand size = 28, antiderivative size = 488 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {22}{9} b^2 d \sqrt {d+c^2 d x^2}-\frac {2 a b c d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 \left (d+c^2 d x^2\right )^{3/2}-\frac {2 b^2 c d x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {2 b c d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b^2 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \] Output:

22/9*b^2*d*(c^2*d*x^2+d)^(1/2)-2*a*b*c*d*x*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1) 
^(1/2)+2/27*b^2*(c^2*d*x^2+d)^(3/2)-2*b^2*c*d*x*(c^2*d*x^2+d)^(1/2)*arcsin 
h(c*x)/(c^2*x^2+1)^(1/2)-2/3*b*c*d*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x) 
)/(c^2*x^2+1)^(1/2)-2/9*b*c^3*d*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)) 
/(c^2*x^2+1)^(1/2)+d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2+1/3*(c^2*d*x 
^2+d)^(3/2)*(a+b*arcsinh(c*x))^2-2*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x) 
)^2*arctanh(c*x+(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)-2*b*d*(c^2*d*x^2+d)^( 
1/2)*(a+b*arcsinh(c*x))*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2 
)+2*b*d*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,c*x+(c^2*x^2+1)^( 
1/2))/(c^2*x^2+1)^(1/2)+2*b^2*d*(c^2*d*x^2+d)^(1/2)*polylog(3,-c*x-(c^2*x^ 
2+1)^(1/2))/(c^2*x^2+1)^(1/2)-2*b^2*d*(c^2*d*x^2+d)^(1/2)*polylog(3,c*x+(c 
^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{3} a^2 d \left (4+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {2 a b d \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+a^2 d^{3/2} \log (c x)-a^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {2 a b d \sqrt {d+c^2 d x^2} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 d \sqrt {d+c^2 d x^2} \left (2 \sqrt {1+c^2 x^2}-2 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2+\text {arcsinh}(c x)^2 \left (\log \left (1-e^{-\text {arcsinh}(c x)}\right )-\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )+2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )+2 \left (\operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 d \sqrt {d+c^2 d x^2} \left (27 \sqrt {1+c^2 x^2} \left (2+\text {arcsinh}(c x)^2\right )+\left (2+9 \text {arcsinh}(c x)^2\right ) \cosh (3 \text {arcsinh}(c x))-6 \text {arcsinh}(c x) (9 c x+\sinh (3 \text {arcsinh}(c x)))\right )}{108 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(a^2*d*(4 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/3 - (2*a*b*d*Sqrt[d + c^2*d*x^2] 
*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x 
^2]) + a^2*d^(3/2)*Log[c*x] - a^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x 
^2]] + (2*a*b*d*Sqrt[d + c^2*d*x^2]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c* 
x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-Ar 
cSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x 
])]))/Sqrt[1 + c^2*x^2] + (b^2*d*Sqrt[d + c^2*d*x^2]*(2*Sqrt[1 + c^2*x^2] 
- 2*c*x*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 + ArcSinh[c*x]^2*( 
Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]) + 2*ArcSinh[c*x]* 
(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]) + 2*(Poly 
Log[3, -E^(-ArcSinh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])])))/Sqrt[1 + c^2 
*x^2] + (b^2*d*Sqrt[d + c^2*d*x^2]*(27*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x] 
^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x 
+ Sinh[3*ArcSinh[c*x]])))/(108*Sqrt[1 + c^2*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.99 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.72, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {6223, 6199, 27, 353, 53, 2009, 6221, 2009, 6231, 3042, 26, 4670, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 6199

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 353

\(\displaystyle d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{6} b c \int \frac {c^2 x^2+3}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 53

\(\displaystyle d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{6} b c \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6221

\(\displaystyle d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6231

\(\displaystyle d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{c x}d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 3042

\(\displaystyle d \left (\frac {\sqrt {c^2 d x^2+d} \int i (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 26

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 4670

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 3011

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \left (-2 i b \left (b \int \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i b \left (b \int \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2720

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \left (-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 7143

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-2 i b \left (b \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i b \left (b \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/3 - (2*b*c*d*Sqrt[d + c^2*d 
*x^2]*(-1/6*(b*c*((4*Sqrt[1 + c^2*x^2])/c^2 + (2*(1 + c^2*x^2)^(3/2))/(3*c 
^2))) + x*(a + b*ArcSinh[c*x]) + (c^2*x^3*(a + b*ArcSinh[c*x]))/3))/(3*Sqr 
t[1 + c^2*x^2]) + d*(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 - (2*b*c*S 
qrt[d + c^2*d*x^2]*(a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/Sqr 
t[1 + c^2*x^2] + (I*Sqrt[d + c^2*d*x^2]*((2*I)*(a + b*ArcSinh[c*x])^2*ArcT 
anh[E^ArcSinh[c*x]] - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSin 
h[c*x]]) + b*PolyLog[3, -E^ArcSinh[c*x]]) + (2*I)*b*(-((a + b*ArcSinh[c*x] 
)*PolyLog[2, E^ArcSinh[c*x]]) + b*PolyLog[3, E^ArcSinh[c*x]])))/Sqrt[1 + c 
^2*x^2])
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

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

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 2009

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

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 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 6199

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; 
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

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 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]
 

rule 7143

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1052\) vs. \(2(481)=962\).

Time = 1.14 (sec) , antiderivative size = 1053, normalized size of antiderivative = 2.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1053\)
parts	\(\text {Expression too large to display}\)	\(1053\)

Input:

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

Output:

-2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(3,x*c+(c^2*x^2+1)^( 
1/2))*d+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(3,-x*c-(c^2* 
x^2+1)^(1/2))*d+68/27*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)+2/3*a*b*(d*( 
c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(x*c)*x^4*c^4+2/27*b^2*(d*(c^2*x^2+ 
1))^(1/2)*d/(c^2*x^2+1)*x^4*c^4+70/27*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2 
+1)*x^2*c^2+b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*ln( 
1-x*c-(c^2*x^2+1)^(1/2))*d-b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arc 
sinh(x*c)^2*ln(1+x*c+(c^2*x^2+1)^(1/2))*d+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2 
*x^2+1)^(1/2)*arcsinh(x*c)*polylog(2,x*c+(c^2*x^2+1)^(1/2))*d-2*b^2*(d*(c^ 
2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*polylog(2,-x*c-(c^2*x^2+1)^ 
(1/2))*d+a^2*d*(c^2*d*x^2+d)^(1/2)+4/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^ 
2+1)*arcsinh(x*c)^2+10/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(x 
*c)*x^2*c^2-a^2*d^(3/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)-2*a*b*(d 
*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-x*c-(c^2*x^2+1)^(1/2))*d+ 
8/3*a*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(x*c)+2*a*b*(d*(c^2*x^2 
+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,x*c+(c^2*x^2+1)^(1/2))*d+1/3*(c^2*d 
*x^2+d)^(3/2)*a^2+1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(x*c) 
^2*x^4*c^4+5/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(x*c)^2*x^2* 
c^2-2/9*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^3*c^3 
-8/3*b^2*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x*c-2/9...
 

Fricas [F]

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

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

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, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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