Optimal. Leaf size=19 \[ \frac {\sec ^2(x)}{2 a}-\frac {\sec (x)}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 8
Rule 30
Rule 2606
Rule 2706
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.74, size = 15, normalized size = 0.79 \[ -\frac {2 \, \cos \relax (x) - 1}{2 \, a \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 15, normalized size = 0.79 \[ -\frac {2 \, \cos \relax (x) - 1}{2 \, a \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 18, normalized size = 0.95 \[ \frac {\frac {1}{2 \cos \relax (x )^{2}}-\frac {1}{\cos \relax (x )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 15, normalized size = 0.79 \[ -\frac {2 \, \cos \relax (x) - 1}{2 \, a \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 13, normalized size = 0.68 \[ -\frac {\cos \relax (x)-\frac {1}{2}}{a\,{\cos \relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan ^{3}{\relax (x )}}{\cos {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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