Integrand size = 12, antiderivative size = 60 \[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=-\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \]
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Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3740, 3557, 371} \[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=-\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),-\cot ^2(c+d x)\right )}{d (n p+1)} \]
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Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left (\cot ^{-n p}(c+d x) \left (b \cot ^p(c+d x)\right )^n\right ) \int \cot ^{n p}(c+d x) \, dx \\ & = -\frac {\left (\cot ^{-n p}(c+d x) \left (b \cot ^p(c+d x)\right )^n\right ) \text {Subst}\left (\int \frac {x^{n p}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=-\frac {\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\cot ^2(c+d x)\right )}{d (1+n p)} \]
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\[\int \left (b \cot \left (d x +c \right )^{p}\right )^{n}d x\]
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\[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=\int { \left (b \cot \left (d x + c\right )^{p}\right )^{n} \,d x } \]
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\[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=\int \left (b \cot ^{p}{\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=\int { \left (b \cot \left (d x + c\right )^{p}\right )^{n} \,d x } \]
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\[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=\int { \left (b \cot \left (d x + c\right )^{p}\right )^{n} \,d x } \]
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Timed out. \[ \int \left (b \cot ^p(c+d x)\right )^n \, dx=\int {\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^p\right )}^n \,d x \]
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