Optimal. Leaf size=115 \[ \frac {\cos (e+f x) \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {3+5 \sin (e+f x)}{4 (1+\sin (e+f x))}\right ) (-3-5 \sin (e+f x))^{-m} \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (a+a \sin (e+f x))^m}{4 f m (1-\sin (e+f x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 115, 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) (-5 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},-m;1-m;\frac {5 \sin (e+f x)+3}{4 (\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)}{4 (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 = 1.60, size = 241, normalized size = 2.10 \begin {gather*} -\frac {4^m \, _2F_1\left (1+m,1+2 m;2 (1+m);\frac {4 \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{2 \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\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 {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 \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\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) ((2-i)-(1-2 i) \cos (e+f x)+(2+i) \sin (e+f x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.18, 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 (-5\,\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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