Optimal. Leaf size=60 \[ -\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\cot ^2(c+d x)\right )}{d (n p+1)} \]
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Rubi [A] time = 0.0397252, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3659, 3476, 364} \[ -\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\cot ^2(c+d x)\right )}{d (n p+1)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \left (b \cot ^p(c+d x)\right )^n \, dx &=\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 ) \operatorname{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 \, _2F_1\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*}
Mathematica [A] time = 0.0516629, size = 58, normalized size = 0.97 \[ -\frac{\cot (c+d x) \left (b \cot ^p(c+d x)\right )^n \text{Hypergeometric2F1}\left (1,\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),-\cot ^2(c+d x)\right )}{d n p+d} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.602, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \cot \left ( dx+c \right ) \right ) ^{p} \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot \left (d x + c\right )^{p}\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \cot \left (d x + c\right )^{p}\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot ^{p}{\left (c + d x \right )}\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot \left (d x + c\right )^{p}\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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