Optimal. Leaf size=62 \[ -\frac {4^{-1-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},1+m;\frac {3}{2};\frac {1-\sin (e+f x)}{4 (1+\sin (e+f x))}\right )}{f (1+\sin (e+f x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.79, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2867, 134}
\begin {gather*} -\frac {2^{-2 m-1} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac {\sin (e+f x)+1}{5 \sin (e+f x)+3}\right )^{\frac {1}{2}-m} (5 \sin (e+f x)+3)^{-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};-\frac {1-\sin (e+f x)}{5 \sin (e+f x)+3}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 2867
Rubi steps
\begin {align*} \int (1+\sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) \text {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (3+5 x)^{-1-m}}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {2^{-1-2 m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};-\frac {1-\sin (e+f x)}{3+5 \sin (e+f x)}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac {1+\sin (e+f x)}{3+5 \sin (e+f x)}\right )^{\frac {1}{2}-m} (3+5 \sin (e+f x))^{-m}}{f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.48, size = 238, normalized size = 3.84 \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 ) (1+\sin (e+f x))^m (1+i \cos (e+f x)+\sin (e+f x)) (3+5 \sin (e+f x))^{-m} \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 = 0.14, size = 0, normalized size = 0.00 \[\int \left (1+\sin \left (f x +e \right )\right )^{m} \left (3+5 \sin \left (f x +e \right )\right )^{-1-m}\, dx\]
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 = -0.02 \begin {gather*} \int \frac {{\left (\sin \left (e+f\,x\right )+1\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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