Optimal. Leaf size=33 \[ -\frac {1}{2} \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(x)}\right )+\frac {1}{2} \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(x)}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5819, 3389,
2211, 2235, 2236} \begin {gather*} \frac {1}{2} \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(x)}\right )-\frac {1}{2} \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5819
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+x^2} \sqrt {\sinh ^{-1}(x)}} \, dx &=\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(x)\right )\right )+\frac {1}{2} \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\sinh ^{-1}(x)\right )\\ &=-\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\sinh ^{-1}(x)}\right )+\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\sinh ^{-1}(x)}\right )\\ &=-\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 34, normalized size = 1.03 \begin {gather*} \frac {1}{2} \left (\frac {\sqrt {-\sinh ^{-1}(x)} \Gamma \left (\frac {1}{2},-\sinh ^{-1}(x)\right )}{\sqrt {\sinh ^{-1}(x)}}+\Gamma \left (\frac {1}{2},\sinh ^{-1}(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {x^{2}+1}\, \sqrt {\arcsinh \left (x \right )}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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} \sqrt {\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 = -0.03 \begin {gather*} \int \frac {x}{\sqrt {\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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