Optimal. Leaf size=29 \[ -\frac {a c \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2815, 2750}
\begin {gather*} -\frac {a c \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2815
Rubi steps
\begin {align*} \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx &=(a c) \int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac {a c \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(29)=58\).
time = 0.19, size = 70, normalized size = 2.41 \begin {gather*} \frac {c \left (-3 \cos \left (e+\frac {f x}{2}\right )+\cos \left (e+\frac {3 f x}{2}\right )\right )}{3 a^2 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs.
\(2(27)=54\).
time = 0.23, size = 56, normalized size = 1.93
method | result | size |
risch | \(\frac {2 c \,{\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 c}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(39\) |
derivativedivides | \(\frac {2 c \left (-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {4}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}\right )}{f \,a^{2}}\) | \(56\) |
default | \(\frac {2 c \left (-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {4}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}\right )}{f \,a^{2}}\) | \(56\) |
norman | \(\frac {-\frac {2 c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 c}{3 a f}-\frac {8 c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs.
\(2 (29) = 58\).
time = 0.32, size = 233, normalized size = 8.03 \begin {gather*} -\frac {2 \, {\left (\frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} - \frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (29) = 58\).
time = 0.31, size = 112, normalized size = 3.86 \begin {gather*} -\frac {c \cos \left (f x + e\right )^{2} - c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) + 2 \, c\right )} \sin \left (f x + e\right ) - 2 \, c}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs.
\(2 (27) = 54\).
time = 1.29, size = 158, normalized size = 5.45 \begin {gather*} \begin {cases} - \frac {6 c \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {2 c}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (- c \sin {\left (e \right )} + c\right )}{\left (a \sin {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 39, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + c\right )}}{3 \, a^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.03, size = 54, normalized size = 1.86 \begin {gather*} \frac {2\,c\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-3\right )}{3\,a^2\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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