Optimal. Leaf size=28 \[ \frac{\tan ^3(a+b x)}{3 b}-\frac{\tan (a+b x)}{b}+x \]
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Rubi [A] time = 0.0160507, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ \frac{\tan ^3(a+b x)}{3 b}-\frac{\tan (a+b x)}{b}+x \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \tan ^4(a+b x) \, dx &=\frac{\tan ^3(a+b x)}{3 b}-\int \tan ^2(a+b x) \, dx\\ &=-\frac{\tan (a+b x)}{b}+\frac{\tan ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x-\frac{\tan (a+b x)}{b}+\frac{\tan ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0094322, size = 38, normalized size = 1.36 \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{\tan ^{-1}(\tan (a+b x))}{b}-\frac{\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 28, normalized size = 1. \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{3}}{3}}-\tan \left ( bx+a \right ) +bx+a \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50201, size = 39, normalized size = 1.39 \begin{align*} \frac{\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6484, size = 115, normalized size = 4.11 \begin{align*} \frac{3 \, b x \cos \left (b x + a\right )^{3} -{\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.69654, size = 39, normalized size = 1.39 \begin{align*} \frac{\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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