Optimal. Leaf size=51 \[ -\frac{(d \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (2,\frac{n+1}{2},\frac{n+3}{2},-\cot ^2(e+f x)\right )}{d f (n+1)} \]
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Rubi [A] time = 0.0491075, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 364} \[ -\frac{(d \cot (e+f x))^{n+1} \, _2F_1\left (2,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 364
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n \sin ^2(e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-d x)^n}{\left (1+x^2\right )^2} \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac{(d \cot (e+f x))^{1+n} \, _2F_1\left (2,\frac{1+n}{2};\frac{3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)}\\ \end{align*}
Mathematica [C] time = 3.07399, size = 509, normalized size = 9.98 \[ -\frac{4 (n-3) \sin \left (\frac{1}{2} (e+f x)\right ) \sin ^2(e+f x) \cos ^3\left (\frac{1}{2} (e+f x)\right ) \left (F_1\left (\frac{1}{2}-\frac{n}{2};-n,2;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-F_1\left (\frac{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right ) (d \cot (e+f x))^n}{f (n-1) \left (2 (n-3) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (\frac{1}{2}-\frac{n}{2};-n,2;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 (n-3) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (\frac{1}{2}-\frac{n}{2};-n,3;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 (\cos (e+f x)-1) \left (n F_1\left (\frac{3}{2}-\frac{n}{2};1-n,2;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-n F_1\left (\frac{3}{2}-\frac{n}{2};1-n,3;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+2 F_1\left (\frac{3}{2}-\frac{n}{2};-n,3;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-3 F_1\left (\frac{3}{2}-\frac{n}{2};-n,4;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.099, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( \sin \left ( fx+e \right ) \right ) ^{2}\, 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 (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{2}\,{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 (\cos \left (f x + e\right )^{2} - 1\right )} \left (d \cot \left (f x + e\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 (d \cot{\left (e + f x \right )}\right )^{n} \sin ^{2}{\left (e + f x \right )}\, 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 (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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