Optimal. Leaf size=83 \[ \frac {2^{1+m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},1+m;\frac {3}{2};\frac {7 (1-\sin (e+f x))}{1+\sin (e+f x)}\right ) (3-4 \sin (e+f x))^{-m} (-3+4 \sin (e+f x))^m}{f (1+\sin (e+f x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 113, normalized size of antiderivative = 1.36, 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 {\sqrt {\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right )}{\sqrt {7} 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 (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx &=\frac {\cos (e+f x) \text {Subst}\left (\int \frac {(3-4 x)^{-1-m} (1+x)^{-\frac {1}{2}+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 {\cos (e+f x) \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {2 (3-4 \sin (e+f x))}{1+\sin (e+f x)}\right ) (3-4 \sin (e+f x))^{-m} \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (1+\sin (e+f x))^m}{\sqrt {7} f m (1-\sin (e+f x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(83)=166\).
time = 0.91, size = 176, normalized size = 2.12 \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},\frac {1}{2}-m;\frac {3}{2};\frac {7 \sin ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{3-4 \sin (e+f x)}\right ) (3-4 \sin (e+f x))^{-m} (1+\sin (e+f x))^m \left (\frac {\cos ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{-3+4 \sin (e+f x)}\right )^{\frac {1}{2}-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 = 0.16, size = 0, normalized size = 0.00 \[\int \left (3-4 \sin \left (f x +e \right )\right )^{-1-m} \left (1+\sin \left (f x +e \right )\right )^{m}\, dx\]
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 (\sin \left (e+f\,x\right )+1\right )}^m}{{\left (3-4\,\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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