Integrand size = 26, antiderivative size = 81 \[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 \operatorname {AppellF1}\left (\frac {3}{2},1-n,1,\frac {5}{2},-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4326, 3645, 129, 525, 524} \[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \operatorname {AppellF1}\left (\frac {3}{2},1-n,1,\frac {5}{2},-i \tan (c+d x),i \tan (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]
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Rule 129
Rule 524
Rule 525
Rule 3645
Rule 4326
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^n \, dx \\ & = \frac {\left (i a^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {-\frac {i x}{a}} (a+x)^{-1+n}}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (2 a^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+i a x^2\right )^{-1+n}}{-a^2+i a^2 x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {\left (2 a^2 \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^n\right ) \text {Subst}\left (\int \frac {x^2 \left (1+i x^2\right )^{-1+n}}{-a^2+i a^2 x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {2 \operatorname {AppellF1}\left (\frac {3}{2},1-n,1,\frac {5}{2},-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{3 d \cot ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
\[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx \]
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\[\int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{n}}{\sqrt {\cot \left (d x +c \right )}}d x\]
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\[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]
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