Optimal. Leaf size=22 \[ \frac{\tan ^4(x)}{4}-\frac{\tan ^2(x)}{2}-\log (\cos (x)) \]
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Rubi [A] time = 0.0101662, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3473, 3475} \[ \frac{\tan ^4(x)}{4}-\frac{\tan ^2(x)}{2}-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \tan ^5(x) \, dx &=\frac{\tan ^4(x)}{4}-\int \tan ^3(x) \, dx\\ &=-\frac{1}{2} \tan ^2(x)+\frac{\tan ^4(x)}{4}+\int \tan (x) \, dx\\ &=-\log (\cos (x))-\frac{\tan ^2(x)}{2}+\frac{\tan ^4(x)}{4}\\ \end{align*}
Mathematica [A] time = 0.0030332, size = 20, normalized size = 0.91 \[ \frac{\sec ^4(x)}{4}-\sec ^2(x)-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{4}}{4}}+{\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 = 0.925153, size = 46, normalized size = 2.09 \begin{align*} \frac{4 \, \sin \left (x\right )^{2} - 3}{4 \,{\left (\sin \left (x\right )^{4} - 2 \, \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 = 2.2591, size = 77, normalized size = 3.5 \begin{align*} \frac{1}{4} \, \tan \left (x\right )^{4} - \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 = 0.103746, size = 20, normalized size = 0.91 \begin{align*} - \frac{4 \cos ^{2}{\left (x \right )} - 1}{4 \cos ^{4}{\left (x \right )}} - \log{\left (\cos{\left (x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05882, size = 30, normalized size = 1.36 \begin{align*} \frac{1}{4} \, \tan \left (x\right )^{4} - \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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