Optimal. Leaf size=12 \[ \frac{\tan ^2(x)}{2}+\log (\cos (x)) \]
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Rubi [A] time = 0.0063899, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3473, 3475} \[ \frac{\tan ^2(x)}{2}+\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \tan ^3(x) \, dx &=\frac{\tan ^2(x)}{2}-\int \tan (x) \, dx\\ &=\log (\cos (x))+\frac{\tan ^2(x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0029327, size = 12, normalized size = 1. \[ \frac{\sec ^2(x)}{2}+\log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 17, normalized size = 1.4 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}-{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924025, size = 27, normalized size = 2.25 \begin{align*} -\frac{1}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}} + \frac{1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07571, size = 57, normalized size = 4.75 \begin{align*} \frac{1}{2} \, \tan \left (x\right )^{2} + \frac{1}{2} \, \log \left (\frac{1}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086282, size = 12, normalized size = 1. \begin{align*} \log{\left (\cos{\left (x \right )} \right )} + \frac{1}{2 \cos ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07701, size = 22, normalized size = 1.83 \begin{align*} \frac{1}{2} \, \tan \left (x\right )^{2} - \frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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