Optimal. Leaf size=3 \[ \text {Shi}\left (\sinh ^{-1}(x)\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5819, 3379}
\begin {gather*} \text {Shi}\left (\sinh ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5819
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)} \, dx &=\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(x)\right )\\ &=\text {Shi}\left (\sinh ^{-1}(x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 3, normalized size = 1.00 \begin {gather*} \text {Shi}\left (\sinh ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.08, size = 4, normalized size = 1.33
method | result | size |
default | \(\hyperbolicSineIntegral \left (\arcsinh \left (x \right )\right )\) | \(4\) |
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 {x^{2} + 1} \operatorname {asinh}{\left (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 = -0.33 \begin {gather*} \int \frac {x}{\mathrm {asinh}\left (x\right )\,\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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