Integrand size = 30, antiderivative size = 127 \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {18 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{5 f \sqrt {3+3 \sin (e+f x)}}-\frac {9 \cos (e+f x) \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac {3 \cos (e+f x) (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817} \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f} \]
[In]
[Out]
Rule 2817
Rule 2819
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}+\frac {1}{5} (4 a) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx \\ & = -\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}+\frac {1}{5} \left (2 a^2\right ) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx \\ & = -\frac {2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69 \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {3 \sqrt {3} c^2 \sec ^4(e+f x) (-1+\sin (e+f x))^2 (1+\sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \left (15-10 \sin ^2(e+f x)+3 \sin ^4(e+f x)\right ) \tan (e+f x)}{5 f} \]
[In]
[Out]
Time = 4.64 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.51
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) a^{2} c^{2} \left (3 \left (\cos ^{4}\left (f x +e \right )\right )+4 \left (\cos ^{2}\left (f x +e \right )\right )+8\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{15 f}\) | \(65\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.67 \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {{\left (3 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{2} c^{2}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )} \]
[In]
[Out]
Timed out. \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.23 \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {16 \, {\left (6 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 15 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 10 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}\right )} \sqrt {a} \sqrt {c}}{15 \, f} \]
[In]
[Out]
Time = 1.61 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {a^2\,c^2\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (175\,\sin \left (2\,e+2\,f\,x\right )+28\,\sin \left (4\,e+4\,f\,x\right )+3\,\sin \left (6\,e+6\,f\,x\right )\right )}{240\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
[In]
[Out]