Optimal. Leaf size=22 \[ -x+\frac{\tan ^5(x)}{5}-\frac{\tan ^3(x)}{3}+\tan (x) \]
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Rubi [A] time = 0.0131138, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3473, 8} \[ -x+\frac{\tan ^5(x)}{5}-\frac{\tan ^3(x)}{3}+\tan (x) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tan ^6(x) \, dx &=\frac{\tan ^5(x)}{5}-\int \tan ^4(x) \, dx\\ &=-\frac{1}{3} \tan ^3(x)+\frac{\tan ^5(x)}{5}+\int \tan ^2(x) \, dx\\ &=\tan (x)-\frac{\tan ^3(x)}{3}+\frac{\tan ^5(x)}{5}-\int 1 \, dx\\ &=-x+\tan (x)-\frac{\tan ^3(x)}{3}+\frac{\tan ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.003632, size = 30, normalized size = 1.36 \[ -x+\frac{23 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)-\frac{11}{15} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 19, normalized size = 0.9 \begin{align*} -x+\tan \left ( x \right ) -{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41082, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03569, size = 57, normalized size = 2.59 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067084, size = 31, normalized size = 1.41 \begin{align*} - x + \frac{\sin ^{5}{\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} - \frac{\sin ^{3}{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05793, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} - \frac{1}{3} \, \tan \left (x\right )^{3} - x + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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