Optimal. Leaf size=33 \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(x)}\right )-\frac{1}{2} \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(x)}\right ) \]
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Rubi [A] time = 0.0799689, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {5779, 3308, 2180, 2204, 2205} \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(x)}\right )-\frac{1}{2} \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(x)}\right ) \]
Antiderivative was successfully verified.
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Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{1+x^2} \sqrt{\sinh ^{-1}(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(x)\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(x)\right )\\ &=-\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(x)}\right )+\operatorname{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*}
Mathematica [A] time = 0.068194, size = 34, normalized size = 1.03 \[ \frac{1}{2} \left (\frac{\sqrt{-\sinh ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(x)\right )}{\sqrt{\sinh ^{-1}(x)}}+\text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{{x}^{2}+1}}}{\frac{1}{\sqrt{{\it Arcsinh} \left ( x \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{2} + 1} \sqrt{\operatorname{arsinh}\left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{2} + 1} \sqrt{\operatorname{asinh}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{2} + 1} \sqrt{\operatorname{arsinh}\left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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