Optimal. Leaf size=32 \[ \frac{2 \sqrt{\left (x^2+1\right ) \left (\sqrt{x^2+1}+1\right )}}{\sqrt{x^2+1}} \]
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Rubi [A] time = 0.100574, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6715, 1588} \[ \frac{2 \sqrt{\left (x^2+1\right ) \left (\sqrt{x^2+1}+1\right )}}{\sqrt{x^2+1}} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 1588
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{1+x^2+\left (1+x^2\right )^{3/2}}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x+(1+x)^{3/2}}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{x}{\sqrt{x^2 (1+x)}} \, dx,x,\sqrt{1+x^2}\right )\\ &=\frac{2 \sqrt{\left (1+x^2\right ) \left (1+\sqrt{1+x^2}\right )}}{\sqrt{1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0484645, size = 37, normalized size = 1.16 \[ \frac{2 \left (x^2+\sqrt{x^2+1}+1\right )}{\sqrt{\left (x^2+1\right ) \left (\sqrt{x^2+1}+1\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{1+{x}^{2}+ \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}}}}}\, 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} +{\left (x^{2} + 1\right )}^{\frac{3}{2}} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.555716, size = 66, normalized size = 2.06 \begin{align*} \frac{2 \, \sqrt{x^{2} +{\left (x^{2} + 1\right )}^{\frac{3}{2}} + 1}}{\sqrt{x^{2} + 1}} \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{\left (x^{2} + 1\right ) \left (\sqrt{x^{2} + 1} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11284, size = 20, normalized size = 0.62 \begin{align*} 2 \, \sqrt{\sqrt{x^{2} + 1} + 1} - 2 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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