Integrand size = 13, antiderivative size = 19 \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=-\frac {\sec (x)}{a}+\frac {\sec ^2(x)}{2 a} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2686, 30, 8} \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=\frac {\sec ^2(x)}{2 a}-\frac {\sec (x)}{a} \]
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Rule 8
Rule 30
Rule 2686
Rule 2785
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \sec (x) \tan (x) \, dx}{a}+\frac {\int \sec ^2(x) \tan (x) \, dx}{a} \\ & = -\frac {\text {Subst}(\int 1 \, dx,x,\sec (x))}{a}+\frac {\text {Subst}(\int x \, dx,x,\sec (x))}{a} \\ & = -\frac {\sec (x)}{a}+\frac {\sec ^2(x)}{2 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=\frac {2 \sec ^2(x) \sin ^4\left (\frac {x}{2}\right )}{a} \]
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Time = 0.48 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\frac {1}{2 \cos \left (x \right )^{2}}-\frac {1}{\cos \left (x \right )}}{a}\) | \(18\) |
risch | \(-\frac {2 \left ({\mathrm e}^{3 i x}-{\mathrm e}^{2 i x}+{\mathrm e}^{i x}\right )}{\left ({\mathrm e}^{2 i x}+1\right )^{2} a}\) | \(33\) |
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=-\frac {2 \, \cos \left (x\right ) - 1}{2 \, a \cos \left (x\right )^{2}} \]
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\[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=\frac {\int \frac {\tan ^{3}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \]
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none
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=-\frac {2 \, \cos \left (x\right ) - 1}{2 \, a \cos \left (x\right )^{2}} \]
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none
Time = 0.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=-\frac {2 \, \cos \left (x\right ) - 1}{2 \, a \cos \left (x\right )^{2}} \]
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Time = 13.48 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {\tan ^3(x)}{a+a \cos (x)} \, dx=-\frac {\cos \left (x\right )-\frac {1}{2}}{a\,{\cos \left (x\right )}^2} \]
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