Integrand size = 30, antiderivative size = 41 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {c \cos (e+f x)}{f (3+3 \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {c \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Rule 2817
Rubi steps \begin{align*} \text {integral}& = -\frac {c \cos (e+f x)}{f (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(41)=82\).
Time = 1.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{3 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
[In]
[Out]
Time = 2.85 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{f a \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}\) | \(39\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )} \]
[In]
[Out]
\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {-c \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.51 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {c} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{2 \, a^{\frac {3}{2}} f \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
[In]
[Out]
Time = 7.83 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {2\,\cos \left (e+f\,x\right )\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{a\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}} \]
[In]
[Out]