3.122 \(\int \frac{x^m}{\sinh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{x^m}{\sinh ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable[x^m/ArcSinh[a*x]^2, x]

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Rubi [A]  time = 0.0136401, 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 = {} \[ \int \frac{x^m}{\sinh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m/ArcSinh[a*x]^2,x]

[Out]

Defer[Int][x^m/ArcSinh[a*x]^2, x]

Rubi steps

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

Mathematica [A]  time = 0.410932, size = 0, normalized size = 0. \[ \int \frac{x^m}{\sinh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m/ArcSinh[a*x]^2,x]

[Out]

Integrate[x^m/ArcSinh[a*x]^2, x]

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Maple [A]  time = 0.435, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m} +{\left (a^{3} x^{3} + a x\right )} x^{m}}{{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} + \int \frac{{\left (a^{3}{\left (m + 1\right )} x^{3} + a{\left (m - 1\right )} x\right )}{\left (a^{2} x^{2} + 1\right )} x^{m} +{\left (2 \, a^{4}{\left (m + 1\right )} x^{4} + a^{2}{\left (3 \, m + 1\right )} x^{2} + m\right )} \sqrt{a^{2} x^{2} + 1} x^{m} +{\left (a^{5}{\left (m + 1\right )} x^{5} + 2 \, a^{3}{\left (m + 1\right )} x^{3} + a{\left (m + 1\right )} x\right )} x^{m}}{{\left (a^{5} x^{5} +{\left (a^{2} x^{2} + 1\right )} a^{3} x^{3} + 2 \, a^{3} x^{3} + a x + 2 \,{\left (a^{4} x^{4} + a^{2} x^{2}\right )} \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^2*x^2 + 1)^(3/2)*x^m + (a^3*x^3 + a*x)*x^m)/((a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x
^2 + 1))) + integrate(((a^3*(m + 1)*x^3 + a*(m - 1)*x)*(a^2*x^2 + 1)*x^m + (2*a^4*(m + 1)*x^4 + a^2*(3*m + 1)*
x^2 + m)*sqrt(a^2*x^2 + 1)*x^m + (a^5*(m + 1)*x^5 + 2*a^3*(m + 1)*x^3 + a*(m + 1)*x)*x^m)/((a^5*x^5 + (a^2*x^2
+ 1)*a^3*x^3 + 2*a^3*x^3 + a*x + 2*(a^4*x^4 + a^2*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m/arcsinh(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m/asinh(a*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/arcsinh(a*x)^2, x)