Integrand size = 14, antiderivative size = 32 \[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=-\frac {\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}}}{d} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3740, 3556} \[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=-\frac {\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}}}{d} \]
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Rule 3556
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left (\cot (c+d x) \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}}\right ) \int \tan (c+d x) \, dx \\ & = -\frac {\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}}}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=-\frac {\cot (c+d x) \log (\cos (c+d x)) \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}}}{d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.38 (sec) , antiderivative size = 5979, normalized size of antiderivative = 186.84
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none
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=-\frac {b^{\left (\frac {1}{p}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
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\[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=\int \left (b \tan ^{p}{\left (c + d x \right )}\right )^{\frac {1}{p}}\, dx \]
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\[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=\int { \left (b \tan \left (d x + c\right )^{p}\right )^{\left (\frac {1}{p}\right )} \,d x } \]
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\[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=\int { \left (b \tan \left (d x + c\right )^{p}\right )^{\left (\frac {1}{p}\right )} \,d x } \]
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Timed out. \[ \int \left (b \tan ^p(c+d x)\right )^{\frac {1}{p}} \, dx=\int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^p\right )}^{1/p} \,d x \]
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