Integrand size = 26, antiderivative size = 52 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^2 c f} \]
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Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2751, 3852, 8} \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\frac {2 \tan (e+f x)}{3 a^2 c f}-\frac {\sec (e+f x)}{3 c f \left (a^2 \sin (e+f x)+a^2\right )} \]
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Rule 8
Rule 2751
Rule 2815
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a c} \\ & = -\frac {\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {2 \int \sec ^2(e+f x) \, dx}{3 a^2 c} \\ & = -\frac {\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}-\frac {2 \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a^2 c f} \\ & = -\frac {\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^2 c f} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {10 \cos (e+f x)+4 \cos (2 (e+f x))-8 \sin (e+f x)+5 \sin (2 (e+f x))}{12 a^2 c f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04
method | result | size |
risch | \(-\frac {4 \left (2 \,{\mathrm e}^{i \left (f x +e \right )}+i\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a^{2} c f}\) | \(54\) |
derivativedivides | \(\frac {-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {2}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} c f}\) | \(73\) |
default | \(\frac {-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {2}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} c f}\) | \(73\) |
parallelrisch | \(\frac {2-6 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f \,a^{2} c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(77\) |
norman | \(\frac {-\frac {2 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {2}{3 a c f}-\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(107\) |
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Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {2 \, \cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 1}{3 \, {\left (a^{2} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} c f \cos \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (41) = 82\).
Time = 1.21 (sec) , antiderivative size = 328, normalized size of antiderivative = 6.31 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\begin {cases} - \frac {6 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a^{2} c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c f} - \frac {6 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a^{2} c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c f} - \frac {2 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} c f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a^{2} c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c f} + \frac {2}{3 a^{2} c f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a^{2} c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a^{2} c f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (48) = 96\).
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.73 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\frac {2 \, {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - 1\right )}}{3 \, {\left (a^{2} c + \frac {2 \, a^{2} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a^{2} c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} f} \]
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Time = 0.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {\frac {3}{a^{2} c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}} + \frac {9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7}{a^{2} c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \]
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Time = 6.51 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {2\,\left (3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}{3\,a^2\,c\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \]
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