Integrand size = 30, antiderivative size = 85 \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 i \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {4}{3},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{2/3} \sqrt [6]{1+i \tan (c+d x)}}{2 \sqrt [6]{2} d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3586, 3604, 72, 71} \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 i \sqrt [6]{1+i \tan (c+d x)} (e \sec (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {4}{3},\frac {1}{2} (1-i \tan (c+d x))\right )}{2 \sqrt [6]{2} d \sqrt {a+i a \tan (c+d x)}} \]
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {(e \sec (c+d x))^{2/3} \int \frac {\sqrt [3]{a-i a \tan (c+d x)}}{\sqrt [6]{a+i a \tan (c+d x)}} \, dx}{\sqrt [3]{a-i a \tan (c+d x)} \sqrt [3]{a+i a \tan (c+d x)}} \\ & = \frac {\left (a^2 (e \sec (c+d x))^{2/3}\right ) \text {Subst}\left (\int \frac {1}{(a-i a x)^{2/3} (a+i a x)^{7/6}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [3]{a-i a \tan (c+d x)} \sqrt [3]{a+i a \tan (c+d x)}} \\ & = \frac {\left (a (e \sec (c+d x))^{2/3} \sqrt [6]{\frac {a+i a \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{7/6} (a-i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [6]{2} d \sqrt [3]{a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {3 i \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {4}{3},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{2/3} \sqrt [6]{1+i \tan (c+d x)}}{2 \sqrt [6]{2} d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.36 \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 i \sqrt [6]{2} \left (\frac {e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{2/3} \sqrt [6]{1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{6},\frac {5}{6},-e^{2 i (c+d x)}\right )}{d \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}} \]
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\[\int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}}}{\sqrt {a +i a \tan \left (d x +c \right )}}d x\]
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\[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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\[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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