Integrand size = 21, antiderivative size = 55 \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)} \]
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Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3309, 272, 67} \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=-\frac {\left (a+b \sin ^n(c+d x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)} \]
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Rule 67
Rule 272
Rule 3309
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sin ^n(c+d x)\right )}{d n} \\ & = -\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)} \]
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\[\int \cot \left (d x +c \right ) {\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{p}d x\]
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\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ) \,d x } \]
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\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{p} \cot {\left (c + d x \right )}\, dx \]
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\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ) \,d x } \]
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\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \mathrm {cot}\left (c+d\,x\right )\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]
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