\(\int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx\) [533]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\text {Int}\left (\frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx \]

[In]

Integrate[ArcSinh[a*x]^n/(x^2*Sqrt[1 + a^2*x^2]),x]

[Out]

Integrate[ArcSinh[a*x]^n/(x^2*Sqrt[1 + a^2*x^2]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {\operatorname {arcsinh}\left (a x \right )^{n}}{x^{2} \sqrt {a^{2} x^{2}+1}}d x\]

[In]

int(arcsinh(a*x)^n/x^2/(a^2*x^2+1)^(1/2),x)

[Out]

int(arcsinh(a*x)^n/x^2/(a^2*x^2+1)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x^{2}} \,d x } \]

[In]

integrate(arcsinh(a*x)^n/x^2/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*x^2 + 1)*arcsinh(a*x)^n/(a^2*x^4 + x^2), x)

Sympy [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}^{n}{\left (a x \right )}}{x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx \]

[In]

integrate(asinh(a*x)**n/x**2/(a**2*x**2+1)**(1/2),x)

[Out]

Integral(asinh(a*x)**n/(x**2*sqrt(a**2*x**2 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x^{2}} \,d x } \]

[In]

integrate(arcsinh(a*x)^n/x^2/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^n/(sqrt(a^2*x^2 + 1)*x^2), x)

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x^{2}} \,d x } \]

[In]

integrate(arcsinh(a*x)^n/x^2/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^n/(sqrt(a^2*x^2 + 1)*x^2), x)

Mupad [N/A]

Not integrable

Time = 2.73 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^n}{x^2\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

int(asinh(a*x)^n/(x^2*(a^2*x^2 + 1)^(1/2)),x)

[Out]

int(asinh(a*x)^n/(x^2*(a^2*x^2 + 1)^(1/2)), x)