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

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

[Out]

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

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Rubi [A]  time = 0.0981286, 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 = {} \[ \int \frac{1}{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.131467, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}{\it Arcsinh} \left ( ax \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x^2/arcsinh(a*x)/(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{1}{\sqrt{a^{2} x^{2} + 1} x^{2} \operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a^2*x^2 + 1)*x^2*arcsinh(a*x)), 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}}{{\left (a^{2} x^{4} + x^{2}\right )} \operatorname{arsinh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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