Integrand size = 30, antiderivative size = 179 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\cos (e+f x)}{4 f (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{8 f (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x)}{24 f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{24 c f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2822, 2820, 3855} \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \cos (e+f x) \text {arctanh}(\sin (e+f x))}{8 a^2 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {3 \cos (e+f x)}{8 a^2 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 \cos (e+f x)}{8 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}} \]
[In]
[Out]
Rule 2820
Rule 2822
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}+\frac {3 \int \frac {1}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx}{4 a} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {3 \cos (e+f x)}{8 a f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {3 \cos (e+f x)}{8 a f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {3 \cos (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{8 a^2 c} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {3 \cos (e+f x)}{8 a f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {3 \cos (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {(3 \cos (e+f x)) \int \sec (e+f x) \, dx}{8 a^2 c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {3 \cos (e+f x)}{8 a f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {3 \cos (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {3 \text {arctanh}(\sin (e+f x)) \cos (e+f x)}{8 a^2 c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (154+48 \log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )+6 \cos (2 (e+f x)) \left (31+8 \log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )-8 \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )-48 \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )+3 \left (-9+8 \log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )-8 \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin (e+f x)+69 \sin (3 (e+f x))+24 \log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-24 \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{2304 \sqrt {3} c f (-1+\sin (e+f x)) (1+\sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Time = 3.07 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {\sec \left (f x +e \right ) \left (3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+3 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-3 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )-2 \left (\sin ^{3}\left (f x +e \right )\right )+\sin ^{2}\left (f x +e \right )+5 \sin \left (f x +e \right )\right )}{8 f c \,a^{2} \left (\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}\) | \(200\) |
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.07 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\left [\frac {3 \, {\left (\cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + \cos \left (f x + e\right )^{3}\right )} \sqrt {a c} \log \left (-\frac {a c \cos \left (f x + e\right )^{3} - 2 \, a c \cos \left (f x + e\right ) - 2 \, \sqrt {a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3}}\right ) - 2 \, {\left (3 \, \cos \left (f x + e\right )^{2} - 3 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{16 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )^{3}\right )}}, -\frac {3 \, {\left (\cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + \cos \left (f x + e\right )^{3}\right )} \sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + {\left (3 \, \cos \left (f x + e\right )^{2} - 3 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{8 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )^{3}\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.73 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {c} {\left (\frac {6 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {12 \, \log \left ({\left | \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right )}{a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {6 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{\frac {5}{2}} c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{32 \, f} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]