Integrand size = 30, antiderivative size = 116 \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {i \operatorname {AppellF1}\left (m,\frac {1}{2},1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{2 f m \sqrt {c+d \tan (e+f x)}} \]
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3645, 142, 141} \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {i (a+i a \tan (e+f x))^m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}} \operatorname {AppellF1}\left (m,\frac {1}{2},1,m+1,-\frac {d (i \tan (e+f x)+1)}{i c-d},\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m \sqrt {c+d \tan (e+f x)}} \]
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Rule 141
Rule 142
Rule 3645
Rubi steps \begin{align*} \text {integral}& = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{\left (-a^2+a x\right ) \sqrt {c-\frac {i d x}{a}}} \, dx,x,i a \tan (e+f x)\right )}{f} \\ & = \frac {\left (i a^2 \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{\left (-a^2+a x\right ) \sqrt {\frac {c}{c+i d}-\frac {i d x}{a (c+i d)}}} \, dx,x,i a \tan (e+f x)\right )}{f \sqrt {c+d \tan (e+f x)}} \\ & = -\frac {i \operatorname {AppellF1}\left (m,\frac {1}{2},1,1+m,-\frac {d (1+i \tan (e+f x))}{i c-d},\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{2 f m \sqrt {c+d \tan (e+f x)}} \\ \end{align*}
\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx \]
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\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\sqrt {c +d \tan \left (f x +e \right )}}d x\]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{\sqrt {d \tan \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{\sqrt {d \tan \left (f x + e\right ) + c}} \,d x } \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^m}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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