Integrand size = 36, antiderivative size = 62 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {(2 A+B) \tan (e+f x)}{3 a^2 c f} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 3852, 8} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\frac {(2 A+B) \tan (e+f x)}{3 a^2 c f}-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2 \sin (e+f x)+a^2\right )} \]
[In]
[Out]
Rule 8
Rule 2938
Rule 3046
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{a c} \\ & = -\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {(2 A+B) \int \sec ^2(e+f x) \, dx}{3 a^2 c} \\ & = -\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}-\frac {(2 A+B) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a^2 c f} \\ & = -\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {(2 A+B) \tan (e+f x)}{3 a^2 c f} \\ \end{align*}
Time = 1.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.77 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\frac {\cos (e+f x) (-6 B-2 (A-B) \cos (e+f x)+2 (2 A+B) \cos (2 (e+f x))-8 A \sin (e+f x)-4 B \sin (e+f x)-A \sin (2 (e+f x))+B \sin (2 (e+f x)))}{12 a^2 c f (-1+\sin (e+f x)) (1+\sin (e+f x))^2} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.39
method | result | size |
risch | \(\frac {2 i \left (4 i A \,{\mathrm e}^{i \left (f x +e \right )}+2 i B \,{\mathrm e}^{i \left (f x +e \right )}+3 B \,{\mathrm e}^{2 i \left (f x +e \right )}-2 A -B \right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a^{2} c f}\) | \(86\) |
parallelrisch | \(\frac {-6 A \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-6 A -6 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2 A -4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 A -2 B}{3 f \,a^{2} c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(95\) |
derivativedivides | \(\frac {-\frac {2 \left (\frac {B}{4}+\frac {A}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {-A +B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A -B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {3 A}{4}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} c f}\) | \(97\) |
default | \(\frac {-\frac {2 \left (\frac {B}{4}+\frac {A}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {-A +B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A -B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {3 A}{4}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} c f}\) | \(97\) |
norman | \(\frac {-\frac {2 A +4 B}{6 a c f}-\frac {4 \left (2 A +B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}+\frac {A \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {\left (2 A +4 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a c f}-\frac {\left (8 A +4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a c f}-\frac {\left (14 A +16 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 a c f}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(199\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {{\left (2 \, A + B\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, A + B\right )} \sin \left (f x + e\right ) - A - 2 \, B}{3 \, {\left (a^{2} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} c f \cos \left (f x + e\right )\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (51) = 102\).
Time = 2.22 (sec) , antiderivative size = 578, normalized size of antiderivative = 9.32 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\begin {cases} - \frac {6 A \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 A \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 A \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 A}{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 B \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 {4 B \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 B}{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 + B \sin {\left (e \right )}\right )}{\left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (58) = 116\).
Time = 0.22 (sec) , antiderivative size = 265, normalized size of antiderivative = 4.27 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\frac {2 \, {\left (\frac {B {\left (\frac {2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{2} c + \frac {2 \, a^{2} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a^{2} c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {A {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - 1\right )}}{a^{2} c + \frac {2 \, a^{2} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a^{2} c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}\right )}}{3 \, f} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.56 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=-\frac {\frac {3 \, {\left (A + B\right )}}{a^{2} c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}} + \frac {9 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, A - B}{a^{2} c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \]
[In]
[Out]
Time = 12.71 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.89 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx=\frac {2\,\left (\frac {3\,B}{2}-A\,\cos \left (e+f\,x\right )+B\,\cos \left (e+f\,x\right )+2\,A\,\sin \left (e+f\,x\right )+B\,\sin \left (e+f\,x\right )-A\,\cos \left (2\,e+2\,f\,x\right )-\frac {B\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {A\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {B\,\sin \left (2\,e+2\,f\,x\right )}{2}\right )}{3\,a^2\,c\,f\,\left (2\,\cos \left (e+f\,x\right )+\sin \left (2\,e+2\,f\,x\right )\right )} \]
[In]
[Out]