Integrand size = 37, antiderivative size = 513 \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^3 \sqrt {-a+b} \sqrt {a+b} d^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^3 \sqrt {-a+b} \sqrt {a+b} d^3 f \sqrt {d \sin (e+f x)}}-\frac {4 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 a d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^3 d^4 f \sqrt {\sin (2 e+2 f x)}} \]
-2/5*(g*cos(f*x+e))^(3/2)/a/d/f/g/(d*sin(f*x+e))^(5/2)+2/3*b*(g*cos(f*x+e) )^(3/2)/a^2/d^2/f/g/(d*sin(f*x+e))^(3/2)-4/5*(g*cos(f*x+e))^(3/2)/a/d^3/f/ g/(d*sin(f*x+e))^(1/2)-2*b^2*(g*cos(f*x+e))^(3/2)/a^3/d^3/f/g/(d*sin(f*x+e ))^(1/2)-2*b^3*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)*g^(1/2)*sin(f*x+e)^(1/2)/a^3/d^3/f/ (-a+b)^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2)+2*b^3*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) *g^(1/2)*sin(f*x+e)^(1/2)/a^3/d^3/f/(-a+b)^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e) )^(1/2)+4/5*(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^4/f/sin (2*f*x+2*e)^(1/2)+2*b^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*Elli pticE(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^4/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 = 24.78 (sec) , antiderivative size = 1726, normalized size of antiderivative = 3.36 \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \]
(Sqrt[g*Cos[e + f*x]]*((-2*(2*a^2*Cos[e + f*x] + 5*b^2*Cos[e + f*x])*Csc[e + f*x])/(5*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x])/(3*a^2) - (2*Cot[e + f* x]*Csc[e + f*x]^2)/(5*a))*Sin[e + f*x]^4)/(f*(d*Sin[e + f*x])^(7/2)) - (Sq rt[g*Cos[e + f*x]]*Sin[e + f*x]^(7/2)*((-2*(4*a^3 + 10*a*b^2)*(-(b*AppellF 1[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 - (S qrt[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]] + 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)*Sqr t[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*Sqrt[ 1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[ e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-...
Time = 3.56 (sec) , antiderivative size = 492, normalized size of antiderivative = 0.96, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {3042, 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.
\(\displaystyle \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))}dx\) |
\(\Big \downarrow \) 3389 |
\(\displaystyle \frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{a d}\) |
\(\Big \downarrow \) 3389 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}\) |
\(\Big \downarrow \) 3043 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3389 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3385 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {d \sin (e+f x)}}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {d \sin (e+f x)}}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 3384 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 993 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
\(\Big \downarrow \) 1542 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{5 d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{5 d f g (d \sin (e+f x))^{5/2}}}{a}-\frac {b \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{a d}\) |
-((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]]/(Sqrt[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]]*E llipticE[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*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]])))/(5*d^2))/a
3.15.14.3.1 Defintions of rubi rules used
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. \(2461\) vs. \(2(483)=966\).
Time = 1.87 (sec) , antiderivative size = 2462, normalized size of antiderivative = 4.80
1/15/f*(a-b)*(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)/d^3*(12*(-a^2+b^2)^ (1/2)*2^(1/2)*a^3+30*(-a^2+b^2)^(1/2)*2^(1/2)*a*b^2+15*sec(f*x+e)*(-cot(f* x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot (f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2 )^(1/2)-a),1/2*2^(1/2))*b^4+15*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1) ^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*Ell ipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^( 1/2))*b^3+15*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e )+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+ e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b^3-24*(-a^2 +b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1 /2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1 /2),1/2*2^(1/2))*a^3+12*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)* (-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF( (-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^3-30*(-a^2+b^2)^(1/2)*(-co t(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e) +cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2)) *b^3+15*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)* (-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2) ,-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a*b^3-15*(-cot(f*x+e)+csc(f*x+e...
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx=\int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]