Optimal. Leaf size=46 \[ \frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{f (1+2 m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2821}
\begin {gather*} \frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2821
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{f (1+2 m)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(46)=92\).
time = 0.88, size = 107, normalized size = 2.33 \begin {gather*} \frac {2^{-m} \cos ^{-1-2 m}\left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{2 m} (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{-m} \sin \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{c f (1+2 m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-1-m}\, dx\]
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 [A]
time = 0.35, size = 47, normalized size = 1.02 \begin {gather*} \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \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 [B]
time = 0.41, size = 58, normalized size = 1.26 \begin {gather*} -\frac {\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m}{c\,f\,\left (2\,m+1\right )\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,\left (\sin \left (e+f\,x\right )-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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