Optimal. Leaf size=66 \[ -\frac{i \sin ^{-n}(c+d x) \text{Hypergeometric2F1}\left (1,n,n+1,-\frac{1}{2} i (\cot (c+d x)+i)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]
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Rubi [A] time = 0.0631324, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3083} \[ -\frac{i \sin ^{-n}(c+d x) \, _2F_1\left (1,n;n+1;-\frac{1}{2} i (\cot (c+d x)+i)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]
Antiderivative was successfully verified.
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Rule 3083
Rubi steps
\begin{align*} \int \sin ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx &=-\frac{i \, _2F_1\left (1,n;1+n;-\frac{1}{2} i (i+\cot (c+d x))\right ) \sin ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}\\ \end{align*}
Mathematica [C] time = 3.45183, size = 367, normalized size = 5.56 \[ -\frac{4 \sin \left (\frac{1}{2} (c+d x)\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \sin ^{-n}(c+d x) (a (\cos (c+d x)+i \sin (c+d x)))^n \left (\text{Hypergeometric2F1}\left (1-2 n,1-n,2-n,-i \tan \left (\frac{1}{2} (c+d x)\right )\right )+F_1\left (1-n;-2 n,1;2-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )\right )}{d (n-1) \left (2 F_1\left (1-n;-2 n,1;2-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )+\frac{\left (1-i \tan \left (\frac{1}{2} (c+d x)\right )\right ) \left (-2 n (i \sin (c+d x)+\cos (c+d x)-1) F_1\left (2-n;1-2 n,1;3-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )-(i \sin (c+d x)+\cos (c+d x)-1) F_1\left (2-n;-2 n,2;3-n;-i \tan \left (\frac{1}{2} (c+d x)\right ),i \tan \left (\frac{1}{2} (c+d x)\right )\right )+(n-2) (\cos (c+d x)+1) \left (1+i \tan \left (\frac{1}{2} (c+d x)\right )\right )^{2 n}\right )}{n-2}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.631, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \right ) \right ) ^{n}}{ \left ( \sin \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 (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n} \sin \left (d x + c\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 (\frac{\left (a e^{\left (i \, d x + i \, c\right )}\right )^{n}}{\left (\frac{1}{2} \,{\left (-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-i \, d x - i \, c\right )}\right )^{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n}}{\sin \left (d x + c\right )^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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