Integrand size = 37, antiderivative size = 822 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=-\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {8 g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{15 a d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 a^3 d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^5 d^6 f \sqrt {\sin (2 e+2 f x)}} \]
-2/9*g*(g*cos(f*x+e))^(3/2)/a/d/f/(d*sin(f*x+e))^(9/2)+2/7*b*g*(g*cos(f*x+ e))^(3/2)/a^2/d^2/f/(d*sin(f*x+e))^(7/2)-4/15*g*(g*cos(f*x+e))^(3/2)/a/d^3 /f/(d*sin(f*x+e))^(5/2)+2/5*(a^2-b^2)*g*(g*cos(f*x+e))^(3/2)/a^3/d^3/f/(d* sin(f*x+e))^(5/2)+8/21*b*g*(g*cos(f*x+e))^(3/2)/a^2/d^4/f/(d*sin(f*x+e))^( 3/2)-2/3*b*(a^2-b^2)*g*(g*cos(f*x+e))^(3/2)/a^4/d^4/f/(d*sin(f*x+e))^(3/2) -8/15*g*(g*cos(f*x+e))^(3/2)/a/d^5/f/(d*sin(f*x+e))^(1/2)+4/5*(a^2-b^2)*g* (g*cos(f*x+e))^(3/2)/a^3/d^5/f/(d*sin(f*x+e))^(1/2)+2*b^2*(a^2-b^2)*g*(g*c os(f*x+e))^(3/2)/a^5/d^5/f/(d*sin(f*x+e))^(1/2)-2*b^3*g^(5/2)*EllipticPi(( g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2) ,I)*2^(1/2)*(-a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^(1/2)/a^5/d^5/f/(d*sin(f*x +e))^(1/2)+2*b^3*g^(5/2)*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f* x+e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*(-a+b)^(1/2)*(a+b)^(1/2)*s in(f*x+e)^(1/2)/a^5/d^5/f/(d*sin(f*x+e))^(1/2)+8/15*g^2*(sin(e+1/4*Pi+f*x) ^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f* x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/a/d^6/f/sin(2*f*x+2*e)^(1/2)-4/5*(a^2-b^2 )*g^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi +f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/a^3/d^6/f/sin(2*f *x+2*e)^(1/2)-2*b^2*(a^2-b^2)*g^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi +f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin(f*x +e))^(1/2)/a^5/d^6/f/sin(2*f*x+2*e)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 24.08 (sec) , antiderivative size = 1850, normalized size of antiderivative = 2.25 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \]
((g*Cos[e + f*x])^(5/2)*((2*(2*a^4*Cos[e + f*x] + 9*a^2*b^2*Cos[e + f*x] - 15*b^4*Cos[e + f*x])*Csc[e + f*x])/(15*a^5) - (2*(3*a^2*b*Cos[e + f*x] - 7*b^3*Cos[e + f*x])*Csc[e + f*x]^2)/(21*a^4) + (2*(a^2*Cos[e + f*x] - 3*b^ 2*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x]^ 3)/(7*a^2) - (2*Cot[e + f*x]*Csc[e + f*x]^4)/(9*a))*Sin[e + f*x]^4*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(5/2)*Sin[e + f*x ]^(11/2)*((-2*(4*a^5 + 18*a^3*b^2 - 30*a*b^4)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^4*b + 24*a^2 *b^3 - 30*b^5)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt [2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2] *(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a] *(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Lo g[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^ 2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan [e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan [e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[S in[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^...
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.
| \(\displaystyle \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3378 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2}}dx}{a^2 d}+\frac {g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{11/2}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2}}dx}{a^2 d}+\frac {g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{11/2}}dx}{a}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (\frac {4 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (\frac {4 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \left (\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \left (\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {2 \left (\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{3 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{9 d f g (d \sin (e+f x))^{9/2}}\right )}{a}\) |
3.15.29.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^( m_.), x_Symbol] :> Simp[(a*Sin[e + f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/ (a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] & & NeQ[m, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a *b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Cos[e + f*x])^ n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
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[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int[ (g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] + (-Simp[b*(g^2/(a^2*d) ) Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] - Simp[g^ 2*((a^2 - b^2)/(a^2*d^2)) Int[(g*Cos[e + f*x])^(p - 2)*((d*Sin[e + f*x])^ (n + 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g}, x] && N eQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LeQ[n, -2] || (EqQ [n, -3/2] && EqQ[p, 3/2]))
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(g *Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] - Simp[b/(a*d) Int[(g*Cos[e + f*x])^p*((d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{ a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1 , p, 1] && LtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(4944\) vs. \(2(779)=1558\).
Time = 2.53 (sec) , antiderivative size = 4945, normalized size of antiderivative = 6.02
-1/2520/f*csc(f*x+e)/(d/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot( f*x+e)))^(11/2)*(1-cos(f*x+e))/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)^3*(-g*((1 -cos(f*x+e))^2*csc(f*x+e)^2-1)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(5/2)*(5 040*csc(f*x+e)^4*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4-5040*csc(f*x+e) ^4*a*b^5*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4-35*csc(f*x+e)^10*a^5*b*(-a^2+b^ 2)^(1/2)*(1-cos(f*x+e))^10-90*csc(f*x+e)^9*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f *x+e))^9+90*csc(f*x+e)^9*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^9+49*csc( f*x+e)^8*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^8-35*(-a^2+b^2)^(1/2)*a^5*b +252*csc(f*x+e)^8*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^8-252*csc(f*x+e) ^8*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^8+360*csc(f*x+e)^7*a^5*b*(-a^2+ b^2)^(1/2)*(1-cos(f*x+e))^7-360*csc(f*x+e)^7*a^4*b^2*(-a^2+b^2)^(1/2)*(1-c os(f*x+e))^7-840*csc(f*x+e)^7*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^7+84 0*csc(f*x+e)^7*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^7-686*csc(f*x+e)^6* a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^6+360*csc(f*x+e)^3*a^5*b*(-a^2+b^2)^ (1/2)*(1-cos(f*x+e))^3-360*csc(f*x+e)^3*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f* x+e))^3-840*csc(f*x+e)^3*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^3+840*csc (f*x+e)^3*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^3-90*a^5*b*(-a^2+b^2)^(1 /2)*(csc(f*x+e)-cot(f*x+e))+90*a^4*b^2*(-a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f* x+e))+35*csc(f*x+e)^10*a^6*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^10-49*csc(f*x+e )^8*a^6*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^8+686*csc(f*x+e)^6*a^6*(-a^2+b^...
Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{11/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]