Integrand size = 26, antiderivative size = 53 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a c^2 f} \]
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Time = 0.14 (sec) , antiderivative size = 53, 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} \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {2 \tan (e+f x)}{3 a c^2 f}+\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )} \]
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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)}{c-c \sin (e+f x)} \, dx}{a c} \\ & = \frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {2 \int \sec ^2(e+f x) \, dx}{3 a c^2} \\ & = \frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}-\frac {2 \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a c^2 f} \\ & = \frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a c^2 f} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {10 \cos (e+f x)+4 \cos (2 (e+f x))+8 \sin (e+f x)-5 \sin (2 (e+f x))}{12 a c^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02
method | result | size |
risch | \(\frac {\frac {8 \,{\mathrm e}^{i \left (f x +e \right )}}{3}-\frac {4 i}{3}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a \,c^{2} f}\) | \(54\) |
derivativedivides | \(\frac {-\frac {2}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a \,c^{2} f}\) | \(73\) |
default | \(\frac {-\frac {2}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a \,c^{2} f}\) | \(73\) |
parallelrisch | \(\frac {-2-6 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a \,c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\) | \(77\) |
norman | \(\frac {\frac {2 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2}{3 a c f}-\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a c f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\) | \(107\) |
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Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=-\frac {2 \, \cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1}{3 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (41) = 82\).
Time = 1.18 (sec) , antiderivative size = 328, normalized size of antiderivative = 6.19 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} + \frac {6 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (50) = 100\).
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.68 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {2 \, {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{3 \, {\left (a c^{2} - \frac {2 \, a c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a c^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} f} \]
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Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=-\frac {\frac {3}{a c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7}{a c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}}{6 \, f} \]
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Time = 6.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=-\frac {2\,\left (3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}{3\,a\,c^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^3\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \]
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