Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\sinh ^{-1}(a x)^n}{x \sqrt {a^2 x^2+1}},x\right ) \]
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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 = {} \[ \int \frac {\sinh ^{-1}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx &=\int \frac {\sinh ^{-1}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 5.93, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{-1}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{n}}{a^{2} x^{3} + x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\arcsinh \left (a x \right )^{n}}{x \sqrt {a^{2} x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^n}{x\,\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}^{n}{\left (a x \right )}}{x \sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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