Integrand size = 23, antiderivative size = 65 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {(A-B) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}-\frac {(A+2 B) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )} \]
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Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2829, 2727} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {(A+2 B) \cos (e+f x)}{3 f \left (a^2 \sin (e+f x)+a^2\right )}-\frac {(A-B) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2} \]
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Rule 2727
Rule 2829
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}+\frac {(A+2 B) \int \frac {1}{a+a \sin (e+f x)} \, dx}{3 a} \\ & = -\frac {(A-B) \cos (e+f x)}{3 f (a+a \sin (e+f x))^2}-\frac {(A+2 B) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {\cos (e+f x) (2 A+B+(A+2 B) \sin (e+f x))}{3 a^2 f (1+\sin (e+f x))^2} \]
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Time = 0.47 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {-6 A \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-6 A -6 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4 A -2 B}{3 f \,a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(60\) |
risch | \(-\frac {2 \left (-A +3 i A \,{\mathrm e}^{i \left (f x +e \right )}+3 i B \,{\mathrm e}^{i \left (f x +e \right )}+3 B \,{\mathrm e}^{2 i \left (f x +e \right )}-2 B \right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(68\) |
derivativedivides | \(\frac {-\frac {2 A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (-2 B +2 A \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{2} f}\) | \(70\) |
default | \(\frac {-\frac {2 A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (-2 B +2 A \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 B -2 A}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{a^{2} f}\) | \(70\) |
norman | \(\frac {-\frac {4 A +2 B}{3 a f}-\frac {2 A \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 \left (5 A +B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {\left (2 A +2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {2 \left (A +B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(139\) |
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Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.80 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\frac {{\left (A + 2 \, B\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, A + B\right )} \cos \left (f x + e\right ) + {\left ({\left (A + 2 \, B\right )} \cos \left (f x + e\right ) - A + B\right )} \sin \left (f x + e\right ) + A - B}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (56) = 112\).
Time = 1.12 (sec) , antiderivative size = 372, normalized size of antiderivative = 5.72 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 A \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {6 A \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {4 A}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {6 B \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} - \frac {2 B}{3 a^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\left (e \right )}\right )}{\left (a \sin {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (61) = 122\).
Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.29 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (\frac {A {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {B {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \]
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Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (3 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A + B\right )}}{3 \, a^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} \]
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Time = 13.84 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,A}{2}+\frac {B}{2}-\frac {A\,\cos \left (e+f\,x\right )}{2}+\frac {B\,\cos \left (e+f\,x\right )}{2}+\frac {3\,A\,\sin \left (e+f\,x\right )}{2}+\frac {3\,B\,\sin \left (e+f\,x\right )}{2}\right )}{3\,a^2\,f\,\left (\frac {3\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}-\frac {\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{2}\right )} \]
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