Integrand size = 30, antiderivative size = 85 \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {9 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt {3+3 \sin (e+f x)}}-\frac {3 \cos (e+f x) \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{5 f} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817} \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt {a \sin (e+f x)+a}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 f} \]
[In]
[Out]
Rule 2817
Rule 2819
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{5 f}+\frac {1}{5} (2 a) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx \\ & = -\frac {a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt {a+a \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{5 f} \\ \end{align*}
Time = 3.85 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.75 \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {3 \sqrt {3} c^3 (-1+\sin (e+f x))^3 (1+\sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} (20 \cos (2 (e+f x))+5 \cos (4 (e+f x))+70 \sin (e+f x)+5 \sin (3 (e+f x))-\sin (5 (e+f x)))}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
[In]
[Out]
Time = 2.83 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, c^{3} a \left (2 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-5 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \tan \left (f x +e \right )+5 \sec \left (f x +e \right )\right )}{10 f}\) | \(92\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.19 \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {{\left (5 \, a c^{3} \cos \left (f x + e\right )^{4} - 5 \, a c^{3} - 2 \, {\left (a c^{3} \cos \left (f x + e\right )^{4} - 2 \, a c^{3} \cos \left (f x + e\right )^{2} - 4 \, a c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{10 \, f \cos \left (f x + e\right )} \]
[In]
[Out]
Timed out. \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {8 \, {\left (4 \, a c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 5 \, a c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}\right )} \sqrt {a} \sqrt {c}}{5 \, f} \]
[In]
[Out]
Time = 8.65 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.31 \[ \int (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {a\,c^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (20\,\cos \left (e+f\,x\right )+25\,\cos \left (3\,e+3\,f\,x\right )+5\,\cos \left (5\,e+5\,f\,x\right )+75\,\sin \left (2\,e+2\,f\,x\right )+4\,\sin \left (4\,e+4\,f\,x\right )-\sin \left (6\,e+6\,f\,x\right )\right )}{80\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
[In]
[Out]