Integrand size = 30, antiderivative size = 86 \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 i 2^{2/3} a \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {13}{6},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{7/3} \sqrt [3]{1+i \tan (c+d x)}}{7 d (a+i a \tan (c+d x))^{3/2}} \]
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Time = 0.26 (sec) , antiderivative size = 86, 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} \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 i 2^{2/3} a \sqrt [3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {13}{6},\frac {1}{2} (1-i \tan (c+d x))\right )}{7 d (a+i a \tan (c+d x))^{3/2}} \]
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {(e \sec (c+d x))^{7/3} \int (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{2/3} \, dx}{(a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}} \\ & = \frac {\left (a^2 (e \sec (c+d x))^{7/3}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{a-i a x}}{\sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}} \\ & = \frac {\left (a^2 (e \sec (c+d x))^{7/3} \sqrt [3]{\frac {a+i a \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{a-i a x}}{\sqrt [3]{\frac {1}{2}+\frac {i x}{2}}} \, dx,x,\tan (c+d x)\right )}{\sqrt [3]{2} d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{3/2}} \\ & = \frac {3 i 2^{2/3} a \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {13}{6},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{7/3} \sqrt [3]{1+i \tan (c+d x)}}{7 d (a+i a \tan (c+d x))^{3/2}} \\ \end{align*}
Time = 1.69 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.37 \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {3 i \sqrt [3]{2} e e^{i (c+d x)} \left (\frac {e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{4/3} \left (4+\left (1+e^{2 i (c+d x)}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {5}{3},-e^{2 i (c+d x)}\right )\right )}{5 d \sqrt {a+i a \tan (c+d x)}} \]
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\[\int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {7}{3}}}{\sqrt {a +i a \tan \left (d x +c \right )}}d x\]
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\[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{7/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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