Integrand size = 37, antiderivative size = 598 \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=-\frac {2 (g \cos (e+f x))^{3/2}}{7 a d f g (d \sin (e+f x))^{7/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f g (d \sin (e+f x))^{5/2}}-\frac {8 (g \cos (e+f x))^{3/2}}{21 a d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f g (d \sin (e+f x))^{3/2}}+\frac {4 b (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f g \sqrt {d \sin (e+f x)}}+\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^4 d^4 f g \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^4 \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^4 \sqrt {-a+b} \sqrt {a+b} d^4 f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^4 \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^4 \sqrt {-a+b} \sqrt {a+b} d^4 f \sqrt {d \sin (e+f x)}}+\frac {4 b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 a^2 d^5 f \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^3 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^4 d^5 f \sqrt {\sin (2 e+2 f x)}} \] Output:
-2/7*(g*cos(f*x+e))^(3/2)/a/d/f/g/(d*sin(f*x+e))^(7/2)+2/5*b*(g*cos(f*x+e) )^(3/2)/a^2/d^2/f/g/(d*sin(f*x+e))^(5/2)-8/21*(g*cos(f*x+e))^(3/2)/a/d^3/f /g/(d*sin(f*x+e))^(3/2)-2/3*b^2*(g*cos(f*x+e))^(3/2)/a^3/d^3/f/g/(d*sin(f* x+e))^(3/2)+4/5*b*(g*cos(f*x+e))^(3/2)/a^2/d^4/f/g/(d*sin(f*x+e))^(1/2)+2* b^3*(g*cos(f*x+e))^(3/2)/a^4/d^4/f/g/(d*sin(f*x+e))^(1/2)+2*2^(1/2)*b^4*g^ (1/2)*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)*sin(f*x+e)^(1/2)/a^4/(-a+b)^(1/2)/(a+b)^(1/2)/d^4/f/ (d*sin(f*x+e))^(1/2)-2*2^(1/2)*b^4*g^(1/2)*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)*sin(f*x+e)^(1/2) /a^4/(-a+b)^(1/2)/(a+b)^(1/2)/d^4/f/(d*sin(f*x+e))^(1/2)-4/5*b*(g*cos(f*x+ e))^(1/2)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(d*sin(f*x+e))^(1/2)/a^2/d^ 5/f/sin(2*f*x+2*e)^(1/2)-2*b^3*(g*cos(f*x+e))^(1/2)*EllipticE(cos(e+1/4*Pi +f*x),2^(1/2))*(d*sin(f*x+e))^(1/2)/a^4/d^5/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 = 27.35 (sec) , antiderivative size = 1768, normalized size of antiderivative = 2.96 \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x] )),x]
Output:
(Sqrt[g*Cos[e + f*x]]*((2*(2*a^2*b*Cos[e + f*x] + 5*b^3*Cos[e + f*x])*Csc[ e + f*x])/(5*a^4) - (2*(4*a^2*Cos[e + f*x] + 7*b^2*Cos[e + f*x])*Csc[e + f *x]^2)/(21*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x]^2)/(5*a^2) - (2*Cot[e + f *x]*Csc[e + f*x]^3)/(7*a))*Sin[e + f*x]^5)/(f*(d*Sin[e + f*x])^(9/2)) + (b *Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^(9/2)*((-2*(4*a^3 + 10*a*b^2)*(-(b*Appe llF1[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^2*b + 10*b^3)*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 + (S qrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*S qrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x] ] + 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]]))/(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[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2 *a^2*b - 5*b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sq rt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4,...
Time = 4.48 (sec) , antiderivative size = 581, normalized size of antiderivative = 0.97, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.811, Rules used = {3042, 3389, 3042, 3050, 3042, 3043, 3389, 3042, 3050, 3042, 3050, 3042, 3052, 3042, 3119, 3389, 3042, 3043, 3389, 3042, 3050, 3042, 3052, 3042, 3119, 3385, 3042, 3384, 993, 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.
| \(\displaystyle \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle \frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle \frac {\frac {4 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{7 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a d}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx}{a d}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3389 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3385 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {d \sin (e+f x)}}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {d \sin (e+f x)}}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 3384 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {-\frac {8 (g \cos (e+f x))^{3/2}}{21 d^3 f g (d \sin (e+f x))^{3/2}}-\frac {2 (g \cos (e+f x))^{3/2}}{7 d f g (d \sin (e+f x))^{7/2}}}{a}-\frac {b \left (\frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\right )}{a d}\) |
Input:
Int[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]
Output:
((-2*(g*Cos[e + f*x])^(3/2))/(7*d*f*g*(d*Sin[e + f*x])^(7/2)) - (8*(g*Cos[ e + f*x])^(3/2))/(21*d^3*f*g*(d*Sin[e + f*x])^(3/2)))/a - (b*(-((b*((-2*(g *Cos[e + f*x])^(3/2))/(3*a*d*f*g*(d*Sin[e + f*x])^(3/2)) - (b*((4*Sqrt[2]* b*g*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f* x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(Sqrt[-a + b]*Sqrt[a + b]*Sqrt[ g]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sq rt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(2*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[g]))* Sqrt[Sin[e + f*x]])/(a*d*f*Sqrt[d*Sin[e + f*x]]) + ((-2*(g*Cos[e + f*x])^( 3/2))/(d*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(d^2*f*Sqrt[Sin[2*e + 2*f*x]]))/a))/ (a*d)))/(a*d)) + ((-2*(g*Cos[e + f*x])^(3/2))/(5*d*f*g*(d*Sin[e + f*x])^(5 /2)) + (2*((-2*(g*Cos[e + f*x])^(3/2))/(d*f*g*Sqrt[d*Sin[e + f*x]]) - (2*S qrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(d^ 2*f*Sqrt[Sin[2*e + 2*f*x]])))/(5*d^2))/a))/(a*d)
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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[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[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-4*Sqrt[2]*(g/f) S ubst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g *Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]] *((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Sin[e + f* x]]/Sqrt[d*Sin[e + f*x]] Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[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. \(1429\) vs. \(2(508)=1016\).
Time = 3.95 (sec) , antiderivative size = 1430, normalized size of antiderivative = 2.39
Input:
int((g*cos(f*x+e))^(1/2)/(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)/(d*sin(f*x+e))^(1/2)/d^4*((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))*(-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)+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^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+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))+(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*Elli pticPi((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)*E llipticPi((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^4*(105+105*sec(f*x+e))+(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*EllipticPi((cs c(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*cs c(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*b^5*(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)*EllipticE((csc(f*x+e)-cot(f*x+e)...
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x, al gorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(1/2)/(d*sin(f*x+e))**(9/2)/(a+b*sin(f*x+e)),x)
Output:
Timed out
\[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x, al gorithm="maxima")
Output:
integrate(sqrt(g*cos(f*x + e))/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(9/2 )), x)
\[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e)),x, al gorithm="giac")
Output:
integrate(sqrt(g*cos(f*x + e))/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(9/2 )), x)
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{9/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:
int((g*cos(e + f*x))^(1/2)/((d*sin(e + f*x))^(9/2)*(a + b*sin(e + f*x))),x )
Output:
int((g*cos(e + f*x))^(1/2)/((d*sin(e + f*x))^(9/2)*(a + b*sin(e + f*x))), x)
\[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx=\frac {\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}}{\sin \left (f x +e \right )^{6} b +\sin \left (f x +e \right )^{5} a}d x \right )}{d^{5}} \] Input:
int((g*cos(f*x+e))^(1/2)/(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)))/(sin(e + f*x) **6*b + sin(e + f*x)**5*a),x))/d**5