Optimal. Leaf size=55 \[ \frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{f g (m-n)} \]
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Rubi [A] time = 0.17, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2848} \[ \frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{f g (m-n)} \]
Antiderivative was successfully verified.
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Rule 2848
Rubi steps
\begin {align*} \int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx &=\frac {(g \cos (e+f x))^{-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{f g (m-n)}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 55, normalized size = 1.00 \[ \frac {(a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{f g (m-n)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 83, normalized size = 1.51 \[ \frac {\left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right ) e^{\left (2 \, n \log \left (g \cos \left (f x + e\right )\right ) - n \log \left (a \sin \left (f x + e\right ) + a\right ) + n \log \left (\frac {a c}{g^{2}}\right )\right )}}{f m - f n} \]
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 \[ \int \left (g \cos \left (f x + e\right )\right )^{-m - n - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.05, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{-1-m -n} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 145, normalized size = 2.64 \[ \frac {a^{m} c^{n} g^{-m - n - 1} e^{\left (m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - n \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) + 2 \, n \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right ) - m \log \left (-\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - n \log \left (-\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )\right )}}{f {\left (m - n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.07, size = 53, normalized size = 0.96 \[ \frac {{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^n}{f\,g\,{\left (g\,\cos \left (e+f\,x\right )\right )}^{m+n}\,\left (m-n\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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