Optimal. Leaf size=113 \[ \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {3-5 \sin (e+f x)}{1+\sin (e+f x)}\right ) (3-5 \sin (e+f x))^{-m} \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (a+a \sin (e+f x))^m}{4 f m (1-\sin (e+f x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 113, 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) (3-5 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {3-5 \sin (e+f x)}{\sin (e+f x)+1}\right )}{4 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 (3-5 \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 {(3-5 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 {3-5 \sin (e+f x)}{1+\sin (e+f x)}\right ) (3-5 \sin (e+f x))^{-m} \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (a+a \sin (e+f x))^m}{4 f m (1-\sin (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.60, size = 248, normalized size = 2.19 \begin {gather*} -\frac {2^{-1+2 m} \, _2F_1\left (1+m,1+2 m;2 (1+m);\frac {2 \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+2 \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}\right ) (3-5 \sin (e+f x))^{-m} (a (1+\sin (e+f x)))^m (1+i \cos (e+f x)+\sin (e+f x)) \left (\frac {-\cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+2 \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+2 \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}\right )^m (\cosh (m \log (4))-\sinh (m \log (4)))}{f (1+2 m) ((1-2 i)-(2-i) \cos (e+f x)+(1+2 i) \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (3-5 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (3-5\,\sin \left (e+f\,x\right )\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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