Integrand size = 23, antiderivative size = 155 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d} \]
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Time = 0.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4326, 3656, 926, 129, 525, 524} \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=-\frac {\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d}-\frac {\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} \operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d} \]
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Rule 129
Rule 524
Rule 525
Rule 926
Rule 3656
Rule 4326
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{x^{3/2} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {i (a+b x)^n}{2 (i-x) x^{3/2}}+\frac {i (a+b x)^n}{2 x^{3/2} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{(i-x) x^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{x^{3/2} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^n}{x^2 \left (i-x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^n}{x^2 \left (i+x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {\left (i \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 {\left (1+\frac {b x^2}{a}\right )^n}{x^2 \left (i-x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i \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 {\left (1+\frac {b x^2}{a}\right )^n}{x^2 \left (i+x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {\operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},1,-n,\frac {1}{2},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d} \\ \end{align*}
\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx \]
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\[\int \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}d x\]
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\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\text {Timed out} \]
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\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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