Optimal. Leaf size=27 \[ \frac{\tan ^2(a+b x)}{2 b}+\frac{\log (\cos (a+b x))}{b} \]
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Rubi [A] time = 0.0111813, antiderivative size = 27, 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 = {3473, 3475} \[ \frac{\tan ^2(a+b x)}{2 b}+\frac{\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \tan ^3(a+b x) \, dx &=\frac{\tan ^2(a+b x)}{2 b}-\int \tan (a+b x) \, dx\\ &=\frac{\log (\cos (a+b x))}{b}+\frac{\tan ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.024339, size = 25, normalized size = 0.93 \[ \frac{\tan ^2(a+b x)+2 \log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 26, normalized size = 1. \begin{align*}{\frac{\ln \left ( \cos \left ( bx+a \right ) \right ) }{b}}+{\frac{ \left ( \tan \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00059, size = 42, normalized size = 1.56 \begin{align*} -\frac{\frac{1}{\sin \left (b x + a\right )^{2} - 1} - \log \left (\sin \left (b x + a\right )^{2} - 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63733, size = 89, normalized size = 3.3 \begin{align*} \frac{2 \, \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right )\right ) + 1}{2 \, b \cos \left (b x + a\right )^{2}} \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 = 1.18609, size = 57, normalized size = 2.11 \begin{align*} \frac{\log \left (\frac{\cos \left (b x + a\right )^{2}}{b^{2}}\right )}{2 \, b} - \frac{\cos \left (b x + a\right )^{2} - 1}{2 \, b \cos \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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