Optimal. Leaf size=83 \[ -\frac{(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac{\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f} \]
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Rubi [A] time = 0.0685841, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2707, 78, 65} \[ -\frac{(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac{\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 78
Rule 65
Rubi steps
\begin{align*} \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x) (a+x)^{1+m}}{x^3} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac{\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac{(2-m) \operatorname{Subst}\left (\int \frac{(a+x)^{1+m}}{x^2} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=-\frac{\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac{(2-m) \, _2F_1(2,2+m;3+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{2+m}}{2 a^2 f (2+m)}\\ \end{align*}
Mathematica [A] time = 0.188158, size = 68, normalized size = 0.82 \[ -\frac{(\sin (e+f x)+1)^2 (a (\sin (e+f x)+1))^m \left ((m+2) \csc ^2(e+f x)-(m-2) \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)\right )}{2 f (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.268, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{3} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, 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 \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3}\,{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 (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3}, 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{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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