\(\int \frac {\text {arcsinh}(\frac {x}{a})^{3/2}}{(a^2+x^2)^{3/2}} \, dx\) [493]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\frac {x \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2+x^2}}-\frac {3 \sqrt {1+\frac {x^2}{a^2}} \text {Int}\left (\frac {x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{1+\frac {x^2}{a^2}},x\right )}{2 a^3 \sqrt {a^2+x^2}} \]

[Out]

x*arcsinh(x/a)^(3/2)/a^2/(a^2+x^2)^(1/2)-3/2*(1+x^2/a^2)^(1/2)*Unintegrable(x*arcsinh(x/a)^(1/2)/(1+x^2/a^2),x
)/a^3/(a^2+x^2)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

(x*ArcSinh[x/a]^(3/2))/(a^2*Sqrt[a^2 + x^2]) - (3*Sqrt[1 + x^2/a^2]*Defer[Int][(x*Sqrt[ArcSinh[x/a]])/(1 + x^2
/a^2), x])/(2*a^3*Sqrt[a^2 + x^2])

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2+x^2}}-\frac {\left (3 \sqrt {1+\frac {x^2}{a^2}}\right ) \int \frac {x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{1+\frac {x^2}{a^2}} \, dx}{2 a^3 \sqrt {a^2+x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 9.84 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {asinh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\left (a^{2} + x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{{\left (a^{2} + x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{{\left (a^{2} + x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2+x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {asinh}\left (\frac {x}{a}\right )}^{3/2}}{{\left (a^2+x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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