Optimal. Leaf size=39 \[ -\frac {\cos (e+f x) (-3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m}{f} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, 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 = {23, 2727}
\begin {gather*} -\frac {\cos (e+f x) (-3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 2727
Rubi steps
\begin {align*} \int (-3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\left ((-3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^{1+m}\right ) \int \frac {1}{a+a \sin (e+f x)} \, dx\\ &=-\frac {\cos (e+f x) (-3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m}{f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(106\) vs. \(2(39)=78\).
time = 0.50, size = 106, normalized size = 2.72 \begin {gather*} -\frac {2^{-m} 3^{-1-m} \cos \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 (1+m)} (-1-\sin (e+f x))^{-1-m} (a (1+\sin (e+f x)))^m \sin ^{-1-2 m}\left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (-3-3 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 47, normalized size = 1.21 \begin {gather*} \frac {2 \, a^{m}}{{\left (3^{m + 1} \left (-1\right )^{m} + \frac {3^{m + 1} \left (-1\right )^{m} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f} \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 = 1.21 \begin {gather*} \frac {\left (-\frac {1}{3} \, a\right )^{m} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{3 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (- 3 \sin {\left (e + f x \right )} - 3\right )^{- m - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 815 vs.
\(2 (42) = 84\).
time = 0.76, size = 815, normalized size = 20.90 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 52, normalized size = 1.33 \begin {gather*} \frac {{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (-\cos \left (e+f\,x\right )+\sin \left (e+f\,x\right )\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{f\,{\left (-3\,\sin \left (e+f\,x\right )-3\right )}^{m+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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