Integrand size = 28, antiderivative size = 82 \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\frac {3 a \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \tan ^{\frac {2}{3}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d \sqrt {1+i \tan (c+d x)}} \]
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Time = 0.15 (sec) , antiderivative size = 82, 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 = {3645, 129, 525, 524} \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\frac {3 a \tan ^{\frac {2}{3}}(c+d x) \sqrt {a+i a \tan (c+d x)} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-i \tan (c+d x),i \tan (c+d x)\right )}{2 d \sqrt {1+i \tan (c+d x)}} \]
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Rule 129
Rule 524
Rule 525
Rule 3645
Rubi steps \begin{align*} \text {integral}& = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+x}}{\sqrt [3]{-\frac {i x}{a}} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {x \sqrt {a+i a x^3}}{-a^2+i a^2 x^3} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d} \\ & = -\frac {\left (3 a^3 \sqrt {a+i a \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x \sqrt {1+i x^3}}{-a^2+i a^2 x^3} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d \sqrt {1+i \tan (c+d x)}} \\ & = \frac {3 a \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},1,\frac {5}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \tan ^{\frac {2}{3}}(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d \sqrt {1+i \tan (c+d x)}} \\ \end{align*}
\[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx \]
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\[\int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{\tan \left (d x +c \right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}{\sqrt [3]{\tan {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\tan \left (d x + c\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^{3/2}}{\sqrt [3]{\tan (c+d x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{1/3}} \,d x \]
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