Optimal. Leaf size=54 \[ \frac {x^{1+m} \sinh ^{-1}(a x)^3}{1+m}-\frac {3 a \text {Int}\left (\frac {x^{1+m} \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}},x\right )}{1+m} \]
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Rubi [A]
time = 0.07, 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 = {}
\begin {gather*} \int x^m \sinh ^{-1}(a x)^3 \, dx \end {gather*}
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 = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \sinh ^{-1}(a x)^3 \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 3.41, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{m} \operatorname {asinh}^{3}{\left (a x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^m\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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