Integrand size = 30, antiderivative size = 137 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\cos (e+f x)}{2 f (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x)}{6 f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{6 c f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2822, 2820, 3855} \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\cos (e+f x) \text {arctanh}(\sin (e+f x))}{2 a c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x)}{2 a f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}} \]
[In]
[Out]
Rule 2820
Rule 2822
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{a} \\ & = -\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x)}{2 a f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{2 a c} \\ & = -\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x)}{2 a f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x) \int \sec (e+f x) \, dx}{2 a c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x)}{2 a f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{2 a c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )+\cos (2 (e+f x)) \left (\log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )-\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-2 \sin (e+f x)\right )}{12 \sqrt {3} c f (-1+\sin (e+f x)) (1+\sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Time = 3.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.69
method | result | size |
default | \(-\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \cos \left (f x +e \right )-\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \cos \left (f x +e \right )-\tan \left (f x +e \right )}{2 f a \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, c \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(95\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\left [\frac {\sqrt {a c} \cos \left (f x + e\right )^{3} \log \left (-\frac {a c \cos \left (f x + e\right )^{3} - 2 \, a c \cos \left (f x + e\right ) - 2 \, \sqrt {a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3}}\right ) + 2 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{4 \, a^{2} c^{2} f \cos \left (f x + e\right )^{3}}, -\frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{3} - \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{2 \, a^{2} c^{2} f \cos \left (f x + e\right )^{3}}\right ] \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]