Optimal. Leaf size=11 \[ -\frac{1}{2} \log \left (\cos \left (x^2+1\right )\right ) \]
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Rubi [A] time = 0.010176, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3747, 3475} \[ -\frac{1}{2} \log \left (\cos \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Rule 3747
Rule 3475
Rubi steps
\begin{align*} \int x \tan \left (1+x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \tan (1+x) \, dx,x,x^2\right )\\ &=-\frac{1}{2} \log \left (\cos \left (1+x^2\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0175364, size = 11, normalized size = 1. \[ -\frac{1}{2} \log \left (\cos \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 10, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( \cos \left ({x}^{2}+1 \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963871, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \log \left (\sec \left (x^{2} + 1\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8903, size = 46, normalized size = 4.18 \begin{align*} -\frac{1}{4} \, \log \left (\frac{1}{\tan \left (x^{2} + 1\right )^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.167439, size = 12, normalized size = 1.09 \begin{align*} \frac{\log{\left (\tan ^{2}{\left (x^{2} + 1 \right )} + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11254, size = 14, normalized size = 1.27 \begin{align*} -\frac{1}{2} \, \log \left ({\left | \cos \left (x^{2} + 1\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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