Integrand size = 29, antiderivative size = 44 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \cos (e+f x)}{(c+d) f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2850} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}} \]
[In]
[Out]
Rule 2850
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)}}{(c+d) f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(41)=82\).
Time = 3.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.91
method | result | size |
default | \(-\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (\sin \left (f x +e \right )-1\right ) \left (d \sin \left (f x +e \right )-c \right )}{f \left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right ) \left (c +d \right )}\) | \(84\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (41) = 82\).
Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.93 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{{\left (c d + d^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{2} + c d\right )} f \cos \left (f x + e\right ) - {\left (c^{2} + 2 \, c d + d^{2}\right )} f - {\left ({\left (c d + d^{2}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} + 2 \, c d + d^{2}\right )} f\right )} \sin \left (f x + e\right )} \]
[In]
[Out]
\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (41) = 82\).
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 4.07 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {a} c - \frac {\sqrt {a} {\left (c - 2 \, d\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {a} {\left (c - 2 \, d\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {\sqrt {a} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{{\left (c + d + \frac {{\left (c + d\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {3}{2}} f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (41) = 82\).
Time = 0.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.43 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {4 \, \sqrt {2} {\left (c^{2} d^{2} - 2 \, c d^{3} + d^{4}\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )}{{\left (c^{3} d^{2} - c^{2} d^{3} - c d^{4} + d^{5}\right )} \sqrt {c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + 2 \, c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 6 \, d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + c + d} f} \]
[In]
[Out]
Time = 9.51 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.30 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {4\,\left (2\,c\,\cos \left (e+f\,x\right )+d\,\sin \left (2\,e+2\,f\,x\right )\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{f\,\left (c+d\right )\,\left (4\,c\,d+4\,c^2\,\sin \left (e+f\,x\right )+3\,d^2\,\sin \left (e+f\,x\right )+4\,c^2+2\,d^2-2\,d^2\,\cos \left (2\,e+2\,f\,x\right )-d^2\,\sin \left (3\,e+3\,f\,x\right )+8\,c\,d\,\sin \left (e+f\,x\right )-4\,c\,d\,\cos \left (2\,e+2\,f\,x\right )\right )} \]
[In]
[Out]