Optimal. Leaf size=13 \[ \frac {\sinh ^{-1}(a x)^4}{4 a} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5675} \[ \frac {\sinh ^{-1}(a x)^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sinh ^{-1}(a x)^4}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\sinh ^{-1}(a x)^4}{4 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 23, normalized size = 1.77 \[ \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.92 \[ \frac {\arcsinh \left (a x \right )^{4}}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 11, normalized size = 0.85 \[ \frac {\operatorname {arsinh}\left (a x\right )^{4}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 11, normalized size = 0.85 \[ \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{4\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 10, normalized size = 0.77 \[ \begin {cases} \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{4 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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