Optimal. Leaf size=9 \[ \frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5819, 3379}
\begin {gather*} \frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5819
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 9, normalized size = 1.00 \begin {gather*} \frac {\text {Shi}\left (\sinh ^{-1}(a x)\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.37, size = 10, normalized size = 1.11
method | result | size |
default | \(\frac {\hyperbolicSineIntegral \left (\arcsinh \left (a x \right )\right )}{a^{2}}\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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 [F]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.11 \begin {gather*} \int \frac {x}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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