Integrand size = 24, antiderivative size = 60 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=\frac {a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}+\frac {a \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^3} \]
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Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2815, 2751, 2750} \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=\frac {a \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^3}+\frac {a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4} \]
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Rule 2750
Rule 2751
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^4} \, dx \\ & = \frac {a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}+\frac {1}{5} a \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^3} \, dx \\ & = \frac {a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}+\frac {a \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^3} \\ \end{align*}
Time = 2.70 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.60 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=\frac {a \left (15 \cos \left (e+\frac {f x}{2}\right )-5 \cos \left (e+\frac {3 f x}{2}\right )+5 \sin \left (\frac {f x}{2}\right )+\sin \left (2 e+\frac {5 f x}{2}\right )\right )}{30 c^3 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {2 i a \left (5 i {\mathrm e}^{2 i \left (f x +e \right )}+15 \,{\mathrm e}^{3 i \left (f x +e \right )}+i-5 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 f \,c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5}}\) | \(62\) |
parallelrisch | \(-\frac {2 a \left (15 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-15 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+25 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+4\right )}{15 f \,c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(75\) |
derivativedivides | \(\frac {2 a \left (-\frac {3}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {8}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {14}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{3}}\) | \(86\) |
default | \(\frac {2 a \left (-\frac {3}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {8}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {14}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{3}}\) | \(86\) |
norman | \(\frac {\frac {2 a \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {8 a}{15 c f}-\frac {2 a \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 c f}+\frac {8 a \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {16 a \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {58 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(161\) |
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (58) = 116\).
Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.57 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=-\frac {a \cos \left (f x + e\right )^{3} - 2 \, a \cos \left (f x + e\right )^{2} + 3 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 3 \, a \cos \left (f x + e\right ) + 6 \, a\right )} \sin \left (f x + e\right ) + 6 \, a}{15 \, {\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f - {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (51) = 102\).
Time = 2.38 (sec) , antiderivative size = 571, normalized size of antiderivative = 9.52 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=\begin {cases} - \frac {30 a \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 c^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 c^{3} f} + \frac {30 a \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 c^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 c^{3} f} - \frac {50 a \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 c^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 c^{3} f} + \frac {10 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 c^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 c^{3} f} - \frac {8 a}{15 c^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 c^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 c^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 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. 389 vs. \(2 (58) = 116\).
Time = 0.21 (sec) , antiderivative size = 389, normalized size of antiderivative = 6.48 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (\frac {a {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 7\right )}}{c^{3} - \frac {5 \, c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {10 \, c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, c^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} - \frac {3 \, a {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - 1\right )}}{c^{3} - \frac {5 \, c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {10 \, c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, c^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.32 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (15 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 15 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 25 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, a\right )}}{15 \, c^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}} \]
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Time = 6.86 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.27 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx=\frac {2\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-5\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+25\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-15\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\right )}{15\,c^3\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^5} \]
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