Integrand size = 38, antiderivative size = 180 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\frac {3 (5 A-3 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} a c^{5/2} f}+\frac {3 (5 A-3 B) \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac {(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3046, 2938, 2766, 2729, 2728, 212} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\frac {3 (5 A-3 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} a c^{5/2} f}-\frac {(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {3 (5 A-3 B) \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}} \]
[In]
[Out]
Rule 212
Rule 2728
Rule 2729
Rule 2766
Rule 2938
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx}{a c} \\ & = \frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}+\frac {(5 A-3 B) \int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{8 a c^2} \\ & = \frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac {(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {(3 (5 A-3 B)) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{16 a c} \\ & = \frac {3 (5 A-3 B) \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac {(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {(3 (5 A-3 B)) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{64 a c^2} \\ & = \frac {3 (5 A-3 B) \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac {(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {(3 (5 A-3 B)) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{32 a c^2 f} \\ & = \frac {3 (5 A-3 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} a c^{5/2} f}+\frac {3 (5 A-3 B) \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac {(A+B) \sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}-\frac {(5 A-3 B) \sec (e+f x)}{8 a c^2 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.04 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.24 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (8 (-A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+4 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+(7 A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-(3+3 i) \sqrt [4]{-1} (5 A-3 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+8 (A+B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (7 A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{32 a f (1+\sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(349\) vs. \(2(157)=314\).
Time = 1.01 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {\left (-15 A \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}+30 A \,c^{\frac {5}{2}}+9 B \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}-18 B \,c^{\frac {5}{2}}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (-30 A \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}+40 A \,c^{\frac {5}{2}}+18 B \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}-24 B \,c^{\frac {5}{2}}\right )+30 A \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}-24 A \,c^{\frac {5}{2}}-18 B \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}+40 B \,c^{\frac {5}{2}}}{64 c^{\frac {9}{2}} a \left (\sin \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(350\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.57 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=-\frac {3 \, \sqrt {2} {\left ({\left (5 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{3} + 2 \, {\left (5 \, A - 3 \, B\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, {\left (5 \, A - 3 \, B\right )} \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (3 \, {\left (5 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (5 \, A - 3 \, B\right )} \sin \left (f x + e\right ) - 12 \, A + 20 \, B\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{128 \, {\left (a c^{3} f \cos \left (f x + e\right )^{3} + 2 \, a c^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a c^{3} f \cos \left (f x + e\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {B \sin \left (f x + e\right ) + A}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (157) = 314\).
Time = 0.62 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.80 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\frac {\frac {12 \, \sqrt {2} {\left (5 \, A \sqrt {c} - 3 \, B \sqrt {c}\right )} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (A \sqrt {c} + B \sqrt {c} - \frac {16 \, A \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {90 \, A \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {54 \, B \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}{a c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {128 \, \sqrt {2} {\left (A \sqrt {c} - B \sqrt {c}\right )}}{a c^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\frac {16 \, \sqrt {2} A a c^{\frac {7}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {\sqrt {2} A a c^{\frac {7}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {\sqrt {2} B a c^{\frac {7}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}}{a^{2} c^{6}}}{512 \, f} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
[In]
[Out]