Integrand size = 24, antiderivative size = 29 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {a c \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^3} \]
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Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2815, 2750} \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {a c \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3} \]
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Rule 2750
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (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*}
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(29)=58\).
Time = 1.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\frac {c \left (-3 \cos \left (e+\frac {f x}{2}\right )+\cos \left (e+\frac {3 f x}{2}\right )\right )}{3 a^2 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
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Time = 0.72 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
method | result | size |
parallelrisch | \(\frac {2 c \left (-3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-1\right )}{3 f \,a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(38\) |
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 | \(\frac {2 c \left (\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {4}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{2}}\) | \(56\) |
default | \(\frac {2 c \left (\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {4}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{2}}\) | \(56\) |
norman | \(\frac {-\frac {2 c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2 c}{3 f a}-\frac {8 c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.59 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {c \cos \left (f x + e\right )^{2} - c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) + 2 \, c\right )} \sin \left (f x + e\right ) - 2 \, c}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (27) = 54\).
Time = 1.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.45 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 c \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {2 c}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (- c \sin {\left (e \right )} + c\right )}{\left (a \sin {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (27) = 54\).
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.41 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (\frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} - \frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \]
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Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + c\right )}}{3 \, a^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} \]
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Time = 6.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\frac {2\,c\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-3\right )}{3\,a^2\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^3} \]
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