Optimal. Leaf size=31 \[ -\frac{3 x^2}{4}+\frac{1}{2} x^2 \log \left (x^3+x\right )+\frac{1}{2} \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.0293318, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2525, 444, 43} \[ -\frac{3 x^2}{4}+\frac{1}{2} x^2 \log \left (x^3+x\right )+\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 2525
Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \log \left (x+x^3\right ) \, dx &=\frac{1}{2} x^2 \log \left (x+x^3\right )-\frac{1}{2} \int \frac{x \left (1+3 x^2\right )}{1+x^2} \, dx\\ &=\frac{1}{2} x^2 \log \left (x+x^3\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1+3 x}{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \log \left (x+x^3\right )-\frac{1}{4} \operatorname{Subst}\left (\int \left (3-\frac{2}{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 x^2}{4}+\frac{1}{2} \log \left (1+x^2\right )+\frac{1}{2} x^2 \log \left (x+x^3\right )\\ \end{align*}
Mathematica [A] time = 0.00809, size = 31, normalized size = 1. \[ -\frac{3 x^2}{4}+\frac{1}{2} x^2 \log \left (x^3+x\right )+\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 26, normalized size = 0.8 \begin{align*} -{\frac{3\,{x}^{2}}{4}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{{x}^{2}\ln \left ({x}^{3}+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56524, size = 34, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac{3}{4} \, x^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84842, size = 69, normalized size = 2.23 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac{3}{4} \, x^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.134358, size = 26, normalized size = 0.84 \begin{align*} \frac{x^{2} \log{\left (x^{3} + x \right )}}{2} - \frac{3 x^{2}}{4} + \frac{\log{\left (x^{2} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34124, size = 34, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac{3}{4} \, x^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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