Optimal. Leaf size=88 \[ \frac{c^2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-2 m} \, _2F_1\left (3,-m+n-2;-m+n-1;\frac{1}{2} (1-\sin (e+f x))\right )}{8 f g^5 (m-n+2)} \]
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Rubi [A] time = 0.244216, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2853, 12, 2667, 68} \[ \frac{c^2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-2 m} \, _2F_1\left (3,-m+n-2;-m+n-1;\frac{1}{2} (1-\sin (e+f x))\right )}{8 f g^5 (m-n+2)} \]
Antiderivative was successfully verified.
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Rule 2853
Rule 12
Rule 2667
Rule 68
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{-5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \frac{\sec ^5(e+f x) (c-c \sin (e+f x))^{-m+n}}{g^5} \, dx\\ &=\frac{\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \sec ^5(e+f x) (c-c \sin (e+f x))^{-m+n} \, dx}{g^5}\\ &=-\frac{\left (c^5 (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{(c+x)^{-3-m+n}}{(c-x)^3} \, dx,x,-c \sin (e+f x)\right )}{f g^5}\\ &=\frac{c^2 (g \cos (e+f x))^{-2 m} \, _2F_1\left (3,-2-m+n;-1-m+n;\frac{1}{2} (1-\sin (e+f x))\right ) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n}}{8 f g^5 (2+m-n)}\\ \end{align*}
Mathematica [A] time = 37.122, size = 136, normalized size = 1.55 \[ \frac{\cot ^4\left (\frac{1}{4} (2 e+2 f x-\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )^{n-m} \, _2F_1\left (-m+n-4,-m+n-2;-m+n-1;-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{32 f g^5 (m-n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 11.237, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{-5-2\,m} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \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 (g \cos \left (f x + e\right )\right )^{-2 \, m - 5}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + 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 (\left (g \cos \left (f x + e\right )\right )^{-2 \, m - 5}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\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 \left (g \cos \left (f x + e\right )\right )^{-2 \, m - 5}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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