Optimal. Leaf size=54 \[ \frac {x^{m+1} \sinh ^{-1}(a x)^3}{m+1}-\frac {3 a \text {Int}\left (\frac {x^{m+1} \sinh ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}},x\right )}{m+1} \]
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Rubi [A] time = 0.11, 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 = {} \[ \int x^m \sinh ^{-1}(a x)^3 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \sinh ^{-1}(a x)^3 \, dx &=\frac {x^{1+m} \sinh ^{-1}(a x)^3}{1+m}-\frac {(3 a) \int \frac {x^{1+m} \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 0, normalized size = 0.00 \[ \int x^m \sinh ^{-1}(a x)^3 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arsinh}\left (a x\right )^{3}, x\right ) \]
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 \[ \int x^{m} \operatorname {arsinh}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.19, size = 0, normalized size = 0.00 \[ \int x^{m} \arcsinh \left (a x \right )^{3}\, dx \]
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 \[ \frac {x x^{m} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{m + 1} - \int \frac {3 \, {\left (\sqrt {a^{2} x^{2} + 1} a^{2} x^{2} x^{m} + {\left (a^{3} x^{3} + a x\right )} x^{m}\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{3} {\left (m + 1\right )} x^{3} + a {\left (m + 1\right )} x + {\left (a^{2} {\left (m + 1\right )} x^{2} + m + 1\right )} \sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
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 \[ \int x^{m} \operatorname {asinh}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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