Integrand size = 28, antiderivative size = 110 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{3 c f \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2815, 2766, 2729, 2728, 212} \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Rule 212
Rule 2728
Rule 2729
Rule 2766
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c} \\ & = -\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{2 a} \\ & = \frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{8 a c} \\ & = \frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}-\frac {3 \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 )}{4 a c f} \\ & = \frac {3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\sec (e+f x) \left (1+(3+3 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-3 \sin (e+f x)\right )}{12 c f \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Time = 0.93 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.22
method | result | size |
default | \(-\frac {3 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \sqrt {c \left (\sin \left (f x +e \right )+1\right )}-6 c^{\frac {3}{2}} \sin \left (f x +e \right )+2 c^{\frac {3}{2}}}{8 c^{\frac {5}{2}} a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(134\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (100) = 200\).
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \, \sqrt {2} {\left (\cos \left (f x + e\right ) \sin \left (f x + e\right ) - \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (3 \, \sin \left (f x + e\right ) - 1\right )}}{16 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\int \frac {1}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a} \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (100) = 200\).
Time = 0.33 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.63 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\frac {6 \, \sqrt {2} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{a c^{\frac {3}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (\sqrt {c} + \frac {14 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {3 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )}}{a c^{2} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{32 \, f} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]