3.533 \(\int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{a^2 x^2+1}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.101782, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx &=\int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 1.8322, size = 0, normalized size = 0. \[ \int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.122, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{n}}{{x}^{2}}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{n}}{a^{2} x^{4} + x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{n}{\left (a x \right )}}{x^{2} \sqrt{a^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)