Integrand size = 42, antiderivative size = 357 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {1254 c^5 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {1254 c^4 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 a^2 f g \sqrt {a+a \sin (e+f x)}}+\frac {114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt {a+a \sin (e+f x)}}+\frac {76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}} \]
76/5*c^2*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e) )^(3/2)-4/5*c*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(7/2)/f/g/(a+a*sin(f*x +e))^(5/2)+114/7*c^3*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2)/a^2/f/g/( a+a*sin(f*x+e))^(1/2)+418/5*c^5*(g*cos(f*x+e))^(5/2)/a^2/f/g/(a+a*sin(f*x+ e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+1254/5*c^5*g*(cos(1/2*f*x+1/2*e)^2)^(1/2) /cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2) *(g*cos(f*x+e))^(1/2)/a^2/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+ 1254/35*c^4*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2)/a^2/f/g/(a+a*sin(f *x+e))^(1/2)
Time = 15.40 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {1254 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{9/2}}{5 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{5/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{9/2} \left (\frac {736}{5}+\frac {221}{14} \cos (e+f x)-\frac {1}{14} \cos (3 (e+f x))+\frac {128 \sin \left (\frac {1}{2} (e+f x)\right )}{5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {64}{5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {1472 \sin \left (\frac {1}{2} (e+f x)\right )}{5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {7}{5} \sin (2 (e+f x))\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{5/2}} \]
(1254*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(5*f*Cos[e + f*x]^(3/2)*( Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + ((g *Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5* (c - c*Sin[e + f*x])^(9/2)*(736/5 + (221*Cos[e + f*x])/14 - Cos[3*(e + f*x )]/14 + (128*Sin[(e + f*x)/2])/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) - 64/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) - (1472*Sin[(e + f*x)/2] )/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - (7*Sin[2*(e + f*x)])/5))/(f* (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2))
Time = 2.66 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3329, 3042, 3329, 3042, 3330, 3042, 3330, 3042, 3330, 3042, 3321, 3042, 3121, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle -\frac {19 c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(\sin (e+f x) a+a)^{3/2}}dx}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(\sin (e+f x) a+a)^{3/2}}dx}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {19 c \left (-\frac {15 c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )}{a}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\right )}{5 a}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}-\frac {19 c \left (-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}-\frac {15 c \left (\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {11}{7} c \left (\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}+\frac {7}{5} c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )\right )\right )}{a}\right )}{5 a}\) |
(-4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(5*f*g*(a + a*Sin [e + f*x])^(5/2)) - (19*c*((-4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x ])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)) - (15*c*((2*c*(g*Cos[e + f*x])^ (5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a*Sin[e + f*x]]) + (11*c *((2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a* Sin[e + f*x]]) + (7*c*((2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[ e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos [e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])))/5))/7))/a))/(5*a)
3.2.42.3.1 Defintions of rubi rules used
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* (Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) Int[(g *Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2 *b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] & & EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In tegersQ[2*m, 2*n, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(- b)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(m + n + p))), x] + Simp[a*((2*m + p - 1)/(m + n + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] && !LtQ[0, n, m] && IntegersQ[2 *m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 3.82 (sec) , antiderivative size = 2627, normalized size of antiderivative = 7.36
int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/2),x,m ethod=_RETURNVERBOSE)
-2/35/f*(g*cos(f*x+e))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*g*c^4/(cos(f*x+e)-s in(f*x+e)+1)/(a*(1+sin(f*x+e)))^(1/2)/a^2*(-7189-700*ln(2*(2*(-cos(f*x+e)/ (1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)- cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x +e)*sin(f*x+e)+700*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2 *(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos( f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)*sin(f*x+e)-700*ln(2*(2*(-cos(f*x +e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1 /2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec (f*x+e)*tan(f*x+e)+700*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+ e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(- cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)*tan(f*x+e)-280*cos(f*x+e)^2+ 49*cos(f*x+e)^2*sin(f*x+e)-224*tan(f*x+e)-448*sec(f*x+e)*tan(f*x+e)+224*se c(f*x+e)+1582*cos(f*x+e)-1813*sin(f*x+e)-44*cos(f*x+e)^3-700*ln(2*(2*(-cos (f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2 )^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2) *cos(f*x+e)^2+700*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2* (-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f *x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^2-2800*ln(2*(2*(-cos(f*x+e)/(1+co s(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-co...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.78 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=-\frac {2 \, {\left (5 \, c^{4} g \cos \left (f x + e\right )^{4} - 192 \, c^{4} g \cos \left (f x + e\right )^{2} + 2814 \, c^{4} g + {\left (39 \, c^{4} g \cos \left (f x + e\right )^{2} + 3038 \, c^{4} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 4389 \, {\left (-i \, \sqrt {2} c^{4} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} c^{4} g \sin \left (f x + e\right ) + 2 i \, \sqrt {2} c^{4} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 4389 \, {\left (i \, \sqrt {2} c^{4} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} c^{4} g \sin \left (f x + e\right ) - 2 i \, \sqrt {2} c^{4} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{35 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \sin \left (f x + e\right ) - 2 \, a^{3} f\right )}} \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/ 2),x, algorithm="fricas")
-1/35*(2*(5*c^4*g*cos(f*x + e)^4 - 192*c^4*g*cos(f*x + e)^2 + 2814*c^4*g + (39*c^4*g*cos(f*x + e)^2 + 3038*c^4*g)*sin(f*x + e))*sqrt(g*cos(f*x + e)) *sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) - 4389*(-I*sqrt(2)*c^4 *g*cos(f*x + e)^2 + 2*I*sqrt(2)*c^4*g*sin(f*x + e) + 2*I*sqrt(2)*c^4*g)*sq rt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 4389*(I*sqrt(2)*c^4*g*cos(f*x + e)^2 - 2*I*sqrt(2)*c^4 *g*sin(f*x + e) - 2*I*sqrt(2)*c^4*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, we ierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(a^3*f*cos(f*x + e)^2 - 2*a^3*f*sin(f*x + e) - 2*a^3*f)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/ 2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]