Optimal. Leaf size=81 \[ \frac {(g \cos (e+f x))^{-2 m} \, _2F_1\left (1,-m+n;1-m+n;\frac {1}{2} (1-\sin (e+f x))\right ) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{2 f g (m-n)} \]
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Rubi [A]
time = 0.15, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2932, 12, 2746,
70} \begin {gather*} \frac {(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \, _2F_1\left (1,n-m;-m+n+1;\frac {1}{2} (1-\sin (e+f x))\right )}{2 f g (m-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 70
Rule 2746
Rule 2932
Rubi steps
\begin {align*} \int (g \cos (e+f x))^{-1-2 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 (e+f x) (c-c \sin (e+f x))^{-m+n}}{g} \, 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 (e+f x) (c-c \sin (e+f x))^{-m+n} \, dx}{g}\\ &=-\frac {\left (c (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \text {Subst}\left (\int \frac {(c+x)^{-1-m+n}}{c-x} \, dx,x,-c \sin (e+f x)\right )}{f g}\\ &=\frac {(g \cos (e+f x))^{-2 m} \, _2F_1\left (1,-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))^n}{2 f g (m-n)}\\ \end {align*}
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Mathematica [A]
time = 50.72, size = 115, normalized size = 1.42 \begin {gather*} \frac {(g \cos (e+f x))^{-2 m} \, _2F_1\left (-m+n,-m+n;1-m+n;-\tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )^{-m+n} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^n}{2 f g (m-n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.44, size = 0, normalized size = 0.00 \[\int \left (g \cos \left (f x +e \right )\right )^{-1-2 m} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^n}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{2\,m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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