Optimal. Leaf size=73 \[ -\frac {\sin (e+f x) \sin ^2(e+f x)^{n/2} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2617} \[ -\frac {\sin (e+f x) \sin ^2(e+f x)^{n/2} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n \sin (e+f x) \, dx &=-\frac {(d \cot (e+f x))^{1+n} \, _2F_1\left (\frac {n}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x) \sin ^2(e+f x)^{n/2}}{d f (1+n)}\\ \end {align*}
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Mathematica [C] time = 1.10, size = 264, normalized size = 3.62 \[ -\frac {8 (n-4) \sin ^2\left (\frac {1}{2} (e+f x)\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) F_1\left (1-\frac {n}{2};-n,2;2-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \cot (e+f x))^n}{f (n-2) \left (2 (n-4) \cos ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (1-\frac {n}{2};-n,2;2-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 (\cos (e+f x)-1) \left (n F_1\left (2-\frac {n}{2};1-n,2;3-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 F_1\left (2-\frac {n}{2};-n,3;3-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right ), x\right ) \]
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 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.26, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{n} \sin \left (f x +e \right )\, dx \]
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 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sin \left (e+f\,x\right )\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (d \cot {\left (e + f x \right )}\right )^{n} \sin {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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