Optimal. Leaf size=66 \[ -\frac {i \sin ^{-n}(c+d x) \, _2F_1\left (1,n;n+1;-\frac {1}{2} i (\cot (c+d x)+i)\right ) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, 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*}
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Mathematica [C] time = 3.75, size = 367, normalized size = 5.56 \[ -\frac {4 \sin \left (\frac {1}{2} (c+d x)\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sin ^{-n}(c+d x) (a (\cos (c+d x)+i \sin (c+d x)))^n \left (F_1\left (1-n;-2 n,1;2-n;-i \tan \left (\frac {1}{2} (c+d x)\right ),i \tan \left (\frac {1}{2} (c+d x)\right )\right )+\, _2F_1\left (1-2 n,1-n;2-n;-i \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d (n-1) \left (2 F_1\left (1-n;-2 n,1;2-n;-i \tan \left (\frac {1}{2} (c+d x)\right ),i \tan \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (-2 n (i \sin (c+d x)+\cos (c+d x)-1) F_1\left (2-n;1-2 n,1;3-n;-i \tan \left (\frac {1}{2} (c+d x)\right ),i \tan \left (\frac {1}{2} (c+d x)\right )\right )-(i \sin (c+d x)+\cos (c+d x)-1) F_1\left (2-n;-2 n,2;3-n;-i \tan \left (\frac {1}{2} (c+d x)\right ),i \tan \left (\frac {1}{2} (c+d x)\right )\right )+(n-2) (\cos (c+d x)+1) \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )^{2 n}\right )}{n-2}\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (i \, d n x + i \, c n + n \log \relax (a)\right )}}{\left (\frac {1}{2} \, {\left (-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-i \, d x - i \, c\right )}\right )^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n}}{\sin \left (d x + c\right )^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 10.85, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (d x +c \right )+i a \sin \left (d x +c \right )\right )^{n} \left (\sin ^{-n}\left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{n} \sin \left (d x + c\right )^{-n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a\,\cos \left (c+d\,x\right )+a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\sin \left (c+d\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (i \sin {\left (c + d x \right )} + \cos {\left (c + d x \right )}\right )\right )^{n} \sin ^{- n}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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