Integrand size = 28, antiderivative size = 81 \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {4}{3}}(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.18 (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 = {3645, 129, 525, 524} \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {3 \sqrt {1+i \tan (c+d x)} \tan ^{\frac {4}{3}}(c+d x) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-i \tan (c+d x),i \tan (c+d x)\right )}{4 d \sqrt {a+i a \tan (c+d x)}} \]
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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 [3]{-\frac {i x}{a}}}{(a+x)^{3/2} \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^3}{\left (a+i a x^3\right )^{3/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d} \\ & = -\frac {\left (3 a^2 \sqrt {1+i \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+i x^3\right )^{3/2} \left (-a^2+i a^2 x^3\right )} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{d \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{\frac {4}{3}}(c+d x)}{4 d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx \]
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\[\int \frac {\tan ^{\frac {1}{3}}\left (d x +c \right )}{\sqrt {a +i a \tan \left (d x +c \right )}}d x\]
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Timed out. \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sqrt [3]{\tan {\left (c + d x \right )}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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