Integrand size = 28, antiderivative size = 81 \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=\frac {3 i \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [3]{d \sec (e+f x)} (1+i \tan (e+f x))^{5/6}}{2^{5/6} f (a+i a \tan (e+f x))} \]
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Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3586, 3604, 72, 71} \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=\frac {3 i (1+i \tan (e+f x))^{5/6} \sqrt [3]{d \sec (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{2^{5/6} f (a+i a \tan (e+f x))} \]
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{d \sec (e+f x)} \int \frac {\sqrt [6]{a-i a \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/6}} \, dx}{\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}} \\ & = \frac {\left (a^2 \sqrt [3]{d \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{(a-i a x)^{5/6} (a+i a x)^{11/6}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}} \\ & = \frac {\left (a \sqrt [3]{d \sec (e+f x)} \left (\frac {a+i a \tan (e+f x)}{a}\right )^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{11/6} (a-i a x)^{5/6}} \, dx,x,\tan (e+f x)\right )}{2\ 2^{5/6} f \sqrt [6]{a-i a \tan (e+f x)} (a+i a \tan (e+f x))} \\ & = \frac {3 i \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [3]{d \sec (e+f x)} (1+i \tan (e+f x))^{5/6}}{2^{5/6} f (a+i a \tan (e+f x))} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=-\frac {3 i e^{-2 i (e+f x)} \left (-1-e^{2 i (e+f x)}+4 e^{2 i (e+f x)} \sqrt [3]{1+e^{2 i (e+f x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-e^{2 i (e+f x)}\right )\right ) \sqrt [3]{d \sec (e+f x)}}{10 a f} \]
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\[\int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}{a +i a \tan \left (f x +e \right )}d x\]
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\[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=- \frac {i \int \frac {\sqrt [3]{d \sec {\left (e + f x \right )}}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \]
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Exception generated. \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{d \sec (e+f x)}}{a+i a \tan (e+f x)} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}}{a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]
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