Integrand size = 33, antiderivative size = 173 \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {(A+i B) \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {(A-i B) \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4326, 3684, 3683, 129, 525, 524} \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\frac {(A+i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {(A-i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]
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Rule 129
Rule 524
Rule 525
Rule 3683
Rule 3684
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+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \\ & = \frac {1}{2} \left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int (1+i \tan (c+d x)) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \, dx+\frac {1}{2} \left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int (1-i \tan (c+d x)) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \, dx \\ & = \frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x} (a+b x)^n}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x} (a+b x)^n}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^n}{1-i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^n}{1+i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a}\right )^n}{1-i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a}\right )^n}{1+i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {(A+i B) \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {(A-i B) \operatorname {AppellF1}\left (\frac {3}{2},1,-n,\frac {5}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx \]
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\[\int \frac {\left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )}}d x\]
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\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]
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