Optimal. Leaf size=3 \[ \text {Shi}\left (\sinh ^{-1}(x)\right ) \]
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Rubi [A] time = 0.06, 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 = {5779, 3298} \[ \text {Shi}\left (\sinh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5779
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)} \, dx &=\operatorname {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 = 0.06, size = 3, normalized size = 1.00 \[ \text {Shi}\left (\sinh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 4, normalized size = 1.33 \[ \Shi \left (\arcsinh \relax (x )\right ) \]
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 \[ \int \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)}\,{d x} \]
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 \[ \int \frac {x}{\mathrm {asinh}\relax (x)\,\sqrt {x^2+1}} \,d x \]
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 \[ \int \frac {x}{\sqrt {x^{2} + 1} \operatorname {asinh}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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