Integrand size = 42, antiderivative size = 471 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5525 c^7 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 1.71 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=-\frac {22 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5525 c^7 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac {44 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}} \]
[In]
[Out]
Rule 2719
Rule 2721
Rule 2921
Rule 2929
Rule 2931
Rubi steps \begin{align*} \text {integral}& = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {(3 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{5 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {\left (11 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{35 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {\left (11 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{85 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {\left (33 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{1105 c^4} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {\left (11 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{1105 c^5} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\left (11 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{5525 c^6} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{5525 c^7} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{5525 c^7 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5525 c^7 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5525 c^7 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 13.85 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.42 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=-\frac {22 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{15} (a (1+\sin (e+f x)))^{7/2}}{5525 f \cos ^{\frac {3}{2}}(e+f x) \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 {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{15} \left (\frac {22}{5525}+\frac {32}{25 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{12}}-\frac {416}{175 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}}+\frac {4656}{2975 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}-\frac {2144}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {22}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {22}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{25 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{13}}-\frac {832 \sin \left (\frac {1}{2} (e+f x)\right )}{175 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}}+\frac {9312 \sin \left (\frac {1}{2} (e+f x)\right )}{2975 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {4288 \sin \left (\frac {1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\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}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 5.72 (sec) , antiderivative size = 2170, normalized size of antiderivative = 4.61
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.18 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {2 \, {\left (231 \, a^{3} g \cos \left (f x + e\right )^{6} - 5698 \, a^{3} g \cos \left (f x + e\right )^{4} + 42044 \, a^{3} g \cos \left (f x + e\right )^{2} - 42056 \, a^{3} g + 7 \, {\left (231 \, a^{3} g \cos \left (f x + e\right )^{4} + 1544 \, a^{3} g \cos \left (f x + e\right )^{2} - 4600 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 231 \, {\left (7 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{6} - 56 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 112 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 64 i \, \sqrt {2} a^{3} g + {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{6} + 24 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 80 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 64 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, {\left (-7 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{6} + 56 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 112 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 64 i \, \sqrt {2} a^{3} g + {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{6} - 24 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 80 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 64 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{116025 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{6} - 56 \, c^{8} f \cos \left (f x + e\right )^{4} + 112 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f - {\left (c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f\right )} \sin \left (f x + e\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{15/2}} \,d x \]
[In]
[Out]