Optimal. Leaf size=117 \[ -\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {2 (3-2 \sin (e+f x))}{1+\sin (e+f x)}\right ) \sqrt {-\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (-3+2 \sin (e+f x))^{-m} (a+a \sin (e+f x))^m}{\sqrt {5} f m (1-\sin (e+f x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2867, 134}
\begin {gather*} -\frac {\sqrt {-\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right )}{\sqrt {5} f m (1-\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 2867
Rubi steps
\begin {align*} \int (-3+2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\frac {\left (a^2 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(-3+2 x)^{-1-m} (a+a x)^{-\frac {1}{2}+m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {2 (3-2 \sin (e+f x))}{1+\sin (e+f x)}\right ) \sqrt {-\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (-3+2 \sin (e+f x))^{-m} (a+a \sin (e+f x))^m}{\sqrt {5} f m (1-\sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 155, normalized size = 1.32 \begin {gather*} \frac {2 \cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )^{-\frac {1}{2}+m} \cot \left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \, _2F_1\left (\frac {1}{2},1+m;\frac {3}{2};-5 \tan ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) (a (1+\sin (e+f x)))^m (-3+2 \sin (e+f x))^{-m} \left (-\sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right ) (-3+2 \sin (e+f x))\right )^m \sin ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )^{\frac {1}{2}-m}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (-3+2 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]
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: HeuristicGCDFailed} \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 (2\,\sin \left (e+f\,x\right )-3\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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