\(\int (a^2+x^2)^{3/2} \text {arcsinh}(\frac {x}{a})^{3/2} \, dx\) [123]

Optimal result

Integrand size = 22, antiderivative size = 433 \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=-\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \] Output:

-27/256*a^3*(a^2+x^2)^(1/2)*arcsinh(x/a)^(1/2)/(1+x^2/a^2)^(1/2)-9/32*a*x^ 
2*(a^2+x^2)^(1/2)*arcsinh(x/a)^(1/2)/(1+x^2/a^2)^(1/2)-3/32*(a^2+x^2)^(5/2 
)*arcsinh(x/a)^(1/2)/a/(1+x^2/a^2)^(1/2)+3/8*a^2*x*(a^2+x^2)^(1/2)*arcsinh 
(x/a)^(3/2)+1/4*x*(a^2+x^2)^(3/2)*arcsinh(x/a)^(3/2)+3/20*a^3*(a^2+x^2)^(1 
/2)*arcsinh(x/a)^(5/2)/(1+x^2/a^2)^(1/2)+3/2048*a^3*Pi^(1/2)*(a^2+x^2)^(1/ 
2)*erf(2*arcsinh(x/a)^(1/2))/(1+x^2/a^2)^(1/2)+3/128*a^3*2^(1/2)*Pi^(1/2)* 
(a^2+x^2)^(1/2)*erf(2^(1/2)*arcsinh(x/a)^(1/2))/(1+x^2/a^2)^(1/2)+3/2048*a 
^3*Pi^(1/2)*(a^2+x^2)^(1/2)*erfi(2*arcsinh(x/a)^(1/2))/(1+x^2/a^2)^(1/2)+3 
/128*a^3*2^(1/2)*Pi^(1/2)*(a^2+x^2)^(1/2)*erfi(2^(1/2)*arcsinh(x/a)^(1/2)) 
/(1+x^2/a^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.48 \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {a^3 \sqrt {a^2+x^2} \left (384 \text {arcsinh}\left (\frac {x}{a}\right )^3-480 \text {arcsinh}\left (\frac {x}{a}\right ) \cosh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )+60 \sqrt {2 \pi } \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+60 \sqrt {2 \pi } \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+5 \sqrt {-\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},-4 \text {arcsinh}\left (\frac {x}{a}\right )\right )-5 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},4 \text {arcsinh}\left (\frac {x}{a}\right )\right )+640 \text {arcsinh}\left (\frac {x}{a}\right )^2 \sinh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )\right )}{2560 \sqrt {1+\frac {x^2}{a^2}} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \] Input:

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

Output:

(a^3*Sqrt[a^2 + x^2]*(384*ArcSinh[x/a]^3 - 480*ArcSinh[x/a]*Cosh[2*ArcSinh 
[x/a]] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[x/a]]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] 
+ 60*Sqrt[2*Pi]*Sqrt[ArcSinh[x/a]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 5*Sq 
rt[-ArcSinh[x/a]]*Gamma[5/2, -4*ArcSinh[x/a]] - 5*Sqrt[ArcSinh[x/a]]*Gamma 
[5/2, 4*ArcSinh[x/a]] + 640*ArcSinh[x/a]^2*Sinh[2*ArcSinh[x/a]]))/(2560*Sq 
rt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])
 

Rubi [A] (verified)

Time = 2.98 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.95, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {6201, 27, 6200, 6192, 6198, 6213, 6206, 3042, 3793, 2009, 6234, 3042, 25, 3793, 2009}

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.

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

\(\Big \downarrow \) 6201

\(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}dx-\frac {3 a \sqrt {a^2+x^2} \int \frac {x \left (a^2+x^2\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{a^2}dx}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}dx-\frac {3 \sqrt {a^2+x^2} \int x \left (a^2+x^2\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6200

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \int x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}dx}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \int x \left (a^2+x^2\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \int x \left (a^2+x^2\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \int x \left (a^2+x^2\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^3 \int \frac {\left (\frac {x^2}{a^2}+1\right )^{3/2}}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6206

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \int \frac {\left (\frac {x^2}{a^2}+1\right )^2}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \int \frac {\sin \left (i \text {arcsinh}\left (\frac {x}{a}\right )+\frac {\pi }{2}\right )^4}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \int \left (\frac {\cosh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}+\frac {\cosh \left (4 \text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}+\frac {3}{8 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}\right )d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {x^2}{a^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int -\frac {\sin \left (i \text {arcsinh}\left (\frac {x}{a}\right )\right )^2}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} a^2 \int \frac {\sin \left (i \text {arcsinh}\left (\frac {x}{a}\right )\right )^2}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} a^2 \int \left (\frac {1}{2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}-\frac {\cosh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}\right )d\text {arcsinh}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )-\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} \left (a^2+x^2\right )^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 a \sqrt {\frac {x^2}{a^2}+1}}\)

Input:

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

Output:

(x*(a^2 + x^2)^(3/2)*ArcSinh[x/a]^(3/2))/4 - (3*Sqrt[a^2 + x^2]*(((a^2 + x 
^2)^2*Sqrt[ArcSinh[x/a]])/4 - (a^4*((3*Sqrt[ArcSinh[x/a]])/4 + (Sqrt[Pi]*E 
rf[2*Sqrt[ArcSinh[x/a]]])/32 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] 
)/4 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[x/a]]])/32 + (Sqrt[Pi/2]*Erfi[Sqrt[2]* 
Sqrt[ArcSinh[x/a]]])/4))/8))/(8*a*Sqrt[1 + x^2/a^2]) + (3*a^2*((x*Sqrt[a^2 
 + x^2]*ArcSinh[x/a]^(3/2))/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(5/2))/(5* 
Sqrt[1 + x^2/a^2]) - (3*Sqrt[a^2 + x^2]*((x^2*Sqrt[ArcSinh[x/a]])/2 - (a^2 
*(-Sqrt[ArcSinh[x/a]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/4 + ( 
Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/4))/4))/(4*a*Sqrt[1 + x^2/a^2 
])))/4
 

Defintions of rubi rules used

rule 25

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

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 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6192

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1))   Int 
[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free 
Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[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 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && 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 6206

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Subst[Int 
[x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*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 6234

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a**2 + x**2)*sqrt(asinh(x/a))*asinh(x/a)*x**2,x) + int(sqrt(a**2 
+ x**2)*sqrt(asinh(x/a))*asinh(x/a),x)*a**2