Integrand size = 30, antiderivative size = 178 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2822, 2821} \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]
[In]
[Out]
Rule 2821
Rule 2822
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {3 \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{14 c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{28 c^2} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{280 c^3} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}} \\ \end{align*}
Time = 10.33 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.87 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {8 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3+3 \sin (e+f x))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (3+3 \sin (e+f x))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {6 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (3+3 \sin (e+f x))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (3+3 \sin (e+f x))^{7/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}} \]
[In]
[Out]
Time = 3.45 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{3} \left (13 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-91 \left (\cos ^{5}\left (f x +e \right )\right )-312 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+728 \left (\cos ^{3}\left (f x +e \right )\right )+1075 \sin \left (f x +e \right ) \cos \left (f x +e \right )-1393 \cos \left (f x +e \right )-916 \tan \left (f x +e \right )+756 \sec \left (f x +e \right )\right )}{140 f \left (\cos ^{6}\left (f x +e \right )+6 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-18 \left (\cos ^{4}\left (f x +e \right )\right )-32 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+48 \left (\cos ^{2}\left (f x +e \right )\right )+32 \sin \left (f x +e \right )-32\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{7}}\) | \(200\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {{\left (63 \, a^{3} \cos \left (f x + e\right )^{2} - 76 \, a^{3} + 7 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{140 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) - {\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {{\left (35 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 84 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 70 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 20 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{2240 \, c^{8} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14}} \]
[In]
[Out]
Time = 14.65 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.63 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Too large to display} \]
[In]
[Out]