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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\sinh ^{-1}(a x)^n}{x^2 \sqrt {a^2 x^2+1}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 1.93, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{-1}(a x)^n}{x^2 \sqrt {1+a^2 x^2}} \, dx \]

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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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{n}}{a^{2} x^{4} + x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\arcsinh \left (a x \right )^{n}}{x^{2} \sqrt {a^{2} x^{2}+1}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x^{2}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^n}{x^2\,\sqrt {a^2\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}^{n}{\left (a x \right )}}{x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx \]

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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