Integrand size = 26, antiderivative size = 85 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\frac {\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}+\frac {\sec (e+f x)}{5 a f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{5 a c^3 f} \]
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Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2751, 3852, 8} \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\frac {2 \tan (e+f x)}{5 a c^3 f}+\frac {\sec (e+f x)}{5 a f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2} \]
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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))^2} \, dx}{a c} \\ & = \frac {\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}+\frac {3 \int \frac {\sec ^2(e+f x)}{c-c \sin (e+f x)} \, dx}{5 a c^2} \\ & = \frac {\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}+\frac {\sec (e+f x)}{5 a f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {2 \int \sec ^2(e+f x) \, dx}{5 a c^3} \\ & = \frac {\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}+\frac {\sec (e+f x)}{5 a f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {2 \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{5 a c^3 f} \\ & = \frac {\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}+\frac {\sec (e+f x)}{5 a f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{5 a c^3 f} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\frac {425 \cos (e+f x)+128 \cos (2 (e+f x))-85 \cos (3 (e+f x))+160 \sin (e+f x)-340 \sin (2 (e+f x))-32 \sin (3 (e+f x))}{320 a c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {4 i \left (-4 i {\mathrm e}^{i \left (f x +e \right )}+5 \,{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{5 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a \,c^{3} f}\) | \(66\) |
parallelrisch | \(\frac {-\frac {4}{5}-2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+4 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}}{f a \,c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(90\) |
derivativedivides | \(\frac {-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {4}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {3}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {5}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {7}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a \,c^{3} f}\) | \(103\) |
default | \(\frac {-\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {4}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {3}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {5}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {7}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a \,c^{3} f}\) | \(103\) |
norman | \(\frac {\frac {4 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {4}{5 a c f}-\frac {2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5 a c f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(129\) |
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Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {4 \, \cos \left (f x + e\right )^{2} - {\left (2 \, \cos \left (f x + e\right )^{2} - 3\right )} \sin \left (f x + e\right ) - 2}{5 \, {\left (a c^{3} f \cos \left (f x + e\right )^{3} + 2 \, a c^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a c^{3} f \cos \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (66) = 132\).
Time = 2.36 (sec) , antiderivative size = 614, normalized size of antiderivative = 7.22 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\begin {cases} - \frac {10 \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{5 a c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 20 a c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 25 a c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 25 a c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 20 a c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 5 a c^{3} f} + \frac {20 \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{5 a c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 20 a c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 25 a c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 25 a c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 20 a c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 5 a c^{3} f} - \frac {20 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{5 a c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 20 a c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 25 a c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 25 a c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 20 a c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 5 a c^{3} f} + \frac {6 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{5 a c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 20 a c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 25 a c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 25 a c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 20 a c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 5 a c^{3} f} - \frac {4}{5 a c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 20 a c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 25 a c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 25 a c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 20 a c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 5 a c^{3} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )^{3}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (81) = 162\).
Time = 0.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.48 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {10 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - 2\right )}}{5 \, {\left (a c^{3} - \frac {4 \, a c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, a c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, a c^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {4 \, a c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {a c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} f} \]
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Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {\frac {5}{a c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 90 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 70 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 21}{a c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{20 \, f} \]
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Time = 6.61 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {2\,\left (5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+2\right )}{5\,a\,c^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^5\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \]
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