Optimal. Leaf size=19 \[ \frac{\sec ^2(x)}{2 a}-\frac{\sec (x)}{a} \]
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Rubi [A] time = 0.0596611, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2606, 30, 8} \[ \frac{\sec ^2(x)}{2 a}-\frac{\sec (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2606
Rule 30
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan ^3(x)}{a+a \cos (x)} \, dx &=-\frac{\int \sec (x) \tan (x) \, dx}{a}+\frac{\int \sec ^2(x) \tan (x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\sec (x))}{a}+\frac{\operatorname{Subst}(\int x \, dx,x,\sec (x))}{a}\\ &=-\frac{\sec (x)}{a}+\frac{\sec ^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0224051, size = 17, normalized size = 0.89 \[ \frac{2 \sin ^4\left (\frac{x}{2}\right ) \sec ^2(x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 18, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( - \left ( \cos \left ( x \right ) \right ) ^{-1}+{\frac{1}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14029, size = 20, normalized size = 1.05 \begin{align*} -\frac{2 \, \cos \left (x\right ) - 1}{2 \, a \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29312, size = 46, normalized size = 2.42 \begin{align*} -\frac{2 \, \cos \left (x\right ) - 1}{2 \, a \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan ^{3}{\left (x \right )}}{\cos{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27201, size = 20, normalized size = 1.05 \begin{align*} -\frac{2 \, \cos \left (x\right ) - 1}{2 \, a \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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