Optimal. Leaf size=49 \[ \frac{i (a+i a \cot (c+d x))^n \text{Hypergeometric2F1}\left (1,n,n+1,\frac{1}{2} (1+i \cot (c+d x))\right )}{2 d n} \]
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Rubi [A] time = 0.0279005, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3481, 68} \[ \frac{i (a+i a \cot (c+d x))^n \, _2F_1\left (1,n;n+1;\frac{1}{2} (i \cot (c+d x)+1)\right )}{2 d n} \]
Antiderivative was successfully verified.
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Rule 3481
Rule 68
Rubi steps
\begin{align*} \int (a+i a \cot (c+d x))^n \, dx &=\frac{(i a) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+n}}{a-x} \, dx,x,i a \cot (c+d x)\right )}{d}\\ &=\frac{i (a+i a \cot (c+d x))^n \, _2F_1\left (1,n;1+n;\frac{1}{2} (1+i \cot (c+d x))\right )}{2 d n}\\ \end{align*}
Mathematica [B] time = 0.307475, size = 117, normalized size = 2.39 \[ \frac{i (a+i a \cot (c+d x))^n \left (2 (n+1) \text{Hypergeometric2F1}(1,n,n+1,1+i \cot (c+d x))+(n+i n \cot (c+d x)) \left (\text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{1}{2} (1+i \cot (c+d x))\right )-2 \text{Hypergeometric2F1}(1,n+1,n+2,1+i \cot (c+d x))\right )\right )}{4 d n (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.388, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\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 (i \, a \cot \left (d x + c\right ) + a\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 (-\frac{2 \, a}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}\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 (i a \cot{\left (c + d x \right )} + a\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 (i \, a \cot \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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