\(\int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx\) [85]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\text {Int}\left (\frac {\text {arcsinh}(a x)^{3/2}}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx \]

[In]

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

[Out]

Defer[Int][ArcSinh[a*x]^(3/2)/x, x]

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

Mathematica [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx \]

[In]

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

[Out]

Integrate[ArcSinh[a*x]^(3/2)/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.67 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\int \frac {\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(asinh(a*x)**(3/2)/x, x)

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}{x} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(a*x)^(3/2)/x, x)

Giac [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}}{x} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(a*x)^(3/2)/x, x)

Mupad [N/A]

Not integrable

Time = 2.46 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^{3/2}}{x} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^{3/2}}{x} \,d x \]

[In]

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

[Out]

int(asinh(a*x)^(3/2)/x, x)