Optimal. Leaf size=12 \[ \log \left (\sqrt{x^3+1}+1\right ) \]
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Rubi [A] time = 0.0533348, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {12, 2155, 31} \[ \log \left (\sqrt{x^3+1}+1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2155
Rule 31
Rubi steps
\begin{align*} \int \frac{3 x^2}{2 \left (1+x^3+\sqrt{1+x^3}\right )} \, dx &=\frac{3}{2} \int \frac{x^2}{1+x^3+\sqrt{1+x^3}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{1+x^3}\right )\\ &=\log \left (1+\sqrt{1+x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0213799, size = 12, normalized size = 1. \[ \log \left (\sqrt{x^3+1}+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 39, normalized size = 3.3 \begin{align*}{\frac{3\,\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{2}}+{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46015, size = 54, normalized size = 4.5 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (\frac{x^{3} + \sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1}{\sqrt{x + 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63561, size = 95, normalized size = 7.92 \begin{align*} \frac{3}{2} \, \log \left (x\right ) + \frac{1}{2} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.72642, size = 48, normalized size = 4. \begin{align*} - \frac{\log{\left (2 \sqrt{x^{3} + 1} \right )}}{2} + \frac{\log{\left (2 \sqrt{x^{3} + 1} + 2 \right )}}{2} + \frac{\log{\left (3 x^{3} + 3 \sqrt{x^{3} + 1} + 3 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10482, size = 14, normalized size = 1.17 \begin{align*} \log \left (\sqrt{x^{3} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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