Optimal. Leaf size=51 \[ -\frac{(b \cot (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-\cot ^2(c+d x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.0318807, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3476, 364} \[ -\frac{(b \cot (c+d x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(c+d x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 364
Rubi steps
\begin{align*} \int (b \cot (c+d x))^n \, dx &=-\frac{b \operatorname{Subst}\left (\int \frac{x^n}{b^2+x^2} \, dx,x,b \cot (c+d x)\right )}{d}\\ &=-\frac{(b \cot (c+d x))^{1+n} \, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};-\cot ^2(c+d x)\right )}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0682271, size = 54, normalized size = 1.06 \[ -\frac{\cot (c+d x) (b \cot (c+d x))^n \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+1}{2}+1,-\cot ^2(c+d x)\right )}{d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.378, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cot \left ( dx+c \right ) \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 )\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 )\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{\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 )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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