Integrand size = 37, antiderivative size = 688 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\frac {2 \sqrt {2} b^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a^5 d^{9/2} f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a^5 d^{9/2} f \sqrt {g \cos (e+f x)}}-\frac {2 g \sqrt {g \cos (e+f x)}}{7 a d f (d \sin (e+f x))^{7/2}}+\frac {2 b g \sqrt {g \cos (e+f x)}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac {4 g \sqrt {g \cos (e+f x)}}{7 a d^3 f (d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac {8 b g \sqrt {g \cos (e+f x)}}{5 a^2 d^4 f \sqrt {d \sin (e+f x)}}-\frac {2 b \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}{a^4 d^4 f \sqrt {d \sin (e+f x)}}+\frac {4 g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{7 a d^4 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {2 \left (a^2-b^2\right ) g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{3 a^3 d^4 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {b^2 \left (a^2-b^2\right ) g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a^5 d^4 f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \] Output:
2*2^(1/2)*b^3*(-a^2+b^2)^(1/2)*g^2*cos(f*x+e)^(1/2)*EllipticPi((d*sin(f*x+ e))^(1/2)/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b-(-a^2+b^2)^(1/2)),I)/a^5/d^(9 /2)/f/(g*cos(f*x+e))^(1/2)-2*2^(1/2)*b^3*(-a^2+b^2)^(1/2)*g^2*cos(f*x+e)^( 1/2)*EllipticPi((d*sin(f*x+e))^(1/2)/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b+(- a^2+b^2)^(1/2)),I)/a^5/d^(9/2)/f/(g*cos(f*x+e))^(1/2)-2/7*g*(g*cos(f*x+e)) ^(1/2)/a/d/f/(d*sin(f*x+e))^(7/2)+2/5*b*g*(g*cos(f*x+e))^(1/2)/a^2/d^2/f/( d*sin(f*x+e))^(5/2)-4/7*g*(g*cos(f*x+e))^(1/2)/a/d^3/f/(d*sin(f*x+e))^(3/2 )+2/3*(a^2-b^2)*g*(g*cos(f*x+e))^(1/2)/a^3/d^3/f/(d*sin(f*x+e))^(3/2)+8/5* b*g*(g*cos(f*x+e))^(1/2)/a^2/d^4/f/(d*sin(f*x+e))^(1/2)-2*b*(a^2-b^2)*g*(g *cos(f*x+e))^(1/2)/a^4/d^4/f/(d*sin(f*x+e))^(1/2)+4/7*g^2*InverseJacobiAM( e-1/4*Pi+f*x,2^(1/2))*sin(2*f*x+2*e)^(1/2)/a/d^4/f/(g*cos(f*x+e))^(1/2)/(d *sin(f*x+e))^(1/2)-2/3*(a^2-b^2)*g^2*InverseJacobiAM(e-1/4*Pi+f*x,2^(1/2)) *sin(2*f*x+2*e)^(1/2)/a^3/d^4/f/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)- b^2*(a^2-b^2)*g^2*InverseJacobiAM(e-1/4*Pi+f*x,2^(1/2))*sin(2*f*x+2*e)^(1/ 2)/a^5/d^4/f/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 22.88 (sec) , antiderivative size = 1210, normalized size of antiderivative = 1.76 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f* x])),x]
Output:
((g*Cos[e + f*x])^(3/2)*((-2*b*(a^2 - 5*b^2)*Csc[e + f*x])/(5*a^4) + (2*(a ^2 - 7*b^2)*Csc[e + f*x]^2)/(21*a^3) + (2*b*Csc[e + f*x]^3)/(5*a^2) - (2*C sc[e + f*x]^4)/(7*a))*Sin[e + f*x]^4*Tan[e + f*x])/(f*(d*Sin[e + f*x])^(9/ 2)) - ((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(9/2)*((-2*(2*a^4 + 7*a^2*b^2 - 21*b^4)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF 1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3 *(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^ 2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1 /8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^ 2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqr t[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[S qrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)* Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4) ] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x] ^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sqrt[Sin[e + f*x]])/((1 - Cos[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x])) + (2*(2*a^3*b - 14*a*b^3)*Sqrt[S...
Time = 5.08 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.04, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.811, Rules used = {3042, 3378, 3042, 3050, 3042, 3043, 3050, 3042, 3053, 3042, 3120, 3389, 3042, 3050, 3042, 3053, 3042, 3120, 3389, 3042, 3043, 3389, 3042, 3053, 3042, 3120, 3387, 3042, 3386, 1542}
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.
| \(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3378 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{7/2}}dx}{a^2 d}+\frac {g^2 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{9/2}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{7/2}}dx}{a^2 d}+\frac {g^2 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{9/2}}dx}{a}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (\frac {4 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (\frac {4 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {6 \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {6 \left (\frac {2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{3 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {6 \left (\frac {2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{3 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 d^2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 d^2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{3 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{3 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 d^2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{3 d^2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{a}-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx}{a}-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{a \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{a \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3387 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {g \cos (e+f x)}}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {g \cos (e+f x)}}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 3386 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} d \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \int \frac {1}{\left (\left (b-\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}+\frac {2 \sqrt {2} d \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \int \frac {1}{\left (\left (b+\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}\right )}{a d \sqrt {g \cos (e+f x)}}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \left (\frac {\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {b \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} \sqrt {d} \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (b-\sqrt {b^2-a^2}\right )}+\frac {2 \sqrt {2} \sqrt {d} \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (\sqrt {b^2-a^2}+b\right )}\right )}{a d \sqrt {g \cos (e+f x)}}\right )}{a d}-\frac {2 \sqrt {g \cos (e+f x)}}{a d f g \sqrt {d \sin (e+f x)}}\right )}{a d}\right )}{a^2 d^2}-\frac {b g^2 \left (-\frac {8 \sqrt {g \cos (e+f x)}}{5 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{5 d f g (d \sin (e+f x))^{5/2}}\right )}{a^2 d}+\frac {g^2 \left (\frac {6 \left (\frac {2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{3 d^2 f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)}}{3 d f g (d \sin (e+f x))^{3/2}}\right )}{7 d^2}-\frac {2 \sqrt {g \cos (e+f x)}}{7 d f g (d \sin (e+f x))^{7/2}}\right )}{a}\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x ]
Output:
-((b*g^2*((-2*Sqrt[g*Cos[e + f*x]])/(5*d*f*g*(d*Sin[e + f*x])^(5/2)) - (8* Sqrt[g*Cos[e + f*x]])/(5*d^3*f*g*Sqrt[d*Sin[e + f*x]])))/(a^2*d)) - ((a^2 - b^2)*g^2*(-((b*((-2*Sqrt[g*Cos[e + f*x]])/(a*d*f*g*Sqrt[d*Sin[e + f*x]]) - (b*(-((b*Sqrt[Cos[e + f*x]]*((2*Sqrt[2]*(1 - b/Sqrt[-a^2 + b^2])*Sqrt[d ]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqr t[d]*Sqrt[1 + Cos[e + f*x]])], -1])/((b - Sqrt[-a^2 + b^2])*f) + (2*Sqrt[2 ]*(1 + b/Sqrt[-a^2 + b^2])*Sqrt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/((b + Sqrt[-a^2 + b^2])*f)))/(a*d*Sqrt[g*Cos[e + f*x]])) + (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])))/(a*d)))/(a*d)) + ((-2*Sqrt[g*Cos[e + f*x]])/(3*d*f*g*(d*Sin[e + f*x])^(3/2)) + (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3* d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]))/a))/(a^2*d^2) + (g^2*((- 2*Sqrt[g*Cos[e + f*x]])/(7*d*f*g*(d*Sin[e + f*x])^(7/2)) + (6*((-2*Sqrt[g* Cos[e + f*x]])/(3*d*f*g*(d*Sin[e + f*x])^(3/2)) + (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])))/(7*d^2)))/a
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
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[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[2*Sqrt[2]*d*((b + q)/(f*q)) Subst[Int[1/((d*(b + q) + a*x^2)*Sq rt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Simp[2*Sqrt[2]*d*((b - q)/(f*q)) Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; F reeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.) ]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Cos[e + f *x]]/Sqrt[g*Cos[e + f*x]] Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^ 2, 0]
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. \(1462\) vs. \(2(596)=1192\).
Time = 3.82 (sec) , antiderivative size = 1463, normalized size of antiderivative = 2.13
Input:
int((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x,method=_R ETURNVERBOSE)
Output:
-1/105/f*(g*cos(f*x+e))^(1/2)*(a-b)*g/(d*sin(f*x+e))^(1/2)/d^4*((-a^2+b^2) ^(1/2)*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1 )^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-2*csc(f*x+e)+2*cot(f*x+e) +2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*a*b^3*(-105-105*sec(f*x+e))+(csc( f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+ e)+cot(f*x+e))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^ 2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^4*(-105-105*sec(f*x+e))+(c sc(f*x+e)-cot(f*x+e)+1)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),- a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2) *(-csc(f*x+e)+cot(f*x+e))^(1/2)*a^2*b^3*(105+105*sec(f*x+e))+(csc(f*x+e)-c ot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f *x+e))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^( 1/2)-a),1/2*2^(1/2))*b^5*(-105-105*sec(f*x+e))+(-a^2+b^2)^(1/2)*(csc(f*x+e )-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+co t(f*x+e))^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2 )^(1/2)+a),1/2*2^(1/2))*a*b^3*(-105-105*sec(f*x+e))+(csc(f*x+e)-cot(f*x+e) +1)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2 )+a),1/2*2^(1/2))*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f* x+e))^(1/2)*(-a^2+b^2)^(1/2)*b^4*(-105-105*sec(f*x+e))+(csc(f*x+e)-cot(f*x +e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+...
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x, al gorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)/(d*sin(f*x+e))**(9/2)/(a+b*sin(f*x+e)),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x, al gorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(9 /2)), x)
\[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x, al gorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(9 /2)), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{9/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)/((d*sin(e + f*x))^(9/2)*(a + b*sin(e + f*x))),x )
Output:
int((g*cos(e + f*x))^(3/2)/((d*sin(e + f*x))^(9/2)*(a + b*sin(e + f*x))), x)
\[ \int \frac {(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\frac {\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{6} b +\sin \left (f x +e \right )^{5} a}d x \right ) g}{d^{5}} \] Input:
int((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x)
Output:
(sqrt(g)*sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x))*cos(e + f*x))/ (sin(e + f*x)**6*b + sin(e + f*x)**5*a),x)*g)/d**5