\(\int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx\) [1437]

Optimal result

Integrand size = 37, antiderivative size = 673 \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=-\frac {2 a}{3 \left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a \left (a^2-b^2\right ) d f g^3 (d \sin (e+f x))^{3/2}}+\frac {2 b}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 \left (a^2-b^2\right ) d^2 f g^3 \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^4 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^2 (-a+b)^{3/2} (a+b)^{3/2} d^2 f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^4 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^2 (-a+b)^{3/2} (a+b)^{3/2} d^2 f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 \left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^4 f g \sqrt {g \cos (e+f x)}}+\frac {4 b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^3 f g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^3 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^2 \left (a^2-b^2\right ) d^3 f g^2 \sqrt {\sin (2 e+2 f x)}} \] Output:

-2/3*a/(a^2-b^2)/d/f/g/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(3/2)+2/3*b^2*( 
g*cos(f*x+e))^(3/2)/a/(a^2-b^2)/d/f/g^3/(d*sin(f*x+e))^(3/2)+2*b/(a^2-b^2) 
/d^2/f/g/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)-2*b^3*(g*cos(f*x+e))^(3 
/2)/a^2/(a^2-b^2)/d^2/f/g^3/(d*sin(f*x+e))^(1/2)+2*2^(1/2)*b^4*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^2/(-a+b)^(3/2)/(a+b)^(3/2)/d^2/f/g^(3/2)/(d*sin(f* 
x+e))^(1/2)-2*2^(1/2)*b^4*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^2/(-a+b)^(3/2) 
/(a+b)^(3/2)/d^2/f/g^(3/2)/(d*sin(f*x+e))^(1/2)+8/3*a*(d*sin(f*x+e))^(1/2) 
/(a^2-b^2)/d^3/f/g/(g*cos(f*x+e))^(1/2)-4*b*(d*sin(f*x+e))^(3/2)/(a^2-b^2) 
/d^4/f/g/(g*cos(f*x+e))^(1/2)-4*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-b^2)/d^3/f/g^2/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*s 
in(f*x+e))^(1/2)/a^2/(a^2-b^2)/d^3/f/g^2/sin(2*f*x+2*e)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 27.70 (sec) , antiderivative size = 1727, normalized size of antiderivative = 2.57 \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \] Input:

Integrate[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + 
f*x])),x]
 

Output:

(Cos[e + f*x]^2*Sin[e + f*x]^3*((2*b*Cot[e + f*x])/a^2 - (2*Cot[e + f*x]*C 
sc[e + f*x])/(3*a) + (2*Sec[e + f*x]*(a - b*Sin[e + f*x]))/(a^2 - b^2)))/( 
f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)) - (b*Cos[e + f*x]^(3/2)*S 
in[e + f*x]^(5/2)*((-2*(4*a^3 - 2*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, 
Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 
 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^ 
(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)* 
(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^2*b - 2*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 + (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)*Sqrt[Sin[e + f*x]]*(a + b 
*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^2*b + 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*(-a^2 + b^2)*AppellF1[7/4,...
 

Rubi [A] (verified)

Time = 4.70 (sec) , antiderivative size = 590, normalized size of antiderivative = 0.88, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.757, Rules used = {3042, 3383, 3042, 3317, 3042, 3050, 3042, 3043, 3051, 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.

\(\displaystyle \int \frac {1}{(d \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(d \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2} (a+b \sin (e+f x))}dx\)

\(\Big \downarrow \) 3383

\(\displaystyle \frac {\int \frac {a-b \sin (e+f x)}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}}dx}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a-b \sin (e+f x)}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}}dx}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3317

\(\displaystyle \frac {a \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}}dx-\frac {b \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}dx}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}}dx-\frac {b \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}dx}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3050

\(\displaystyle \frac {a \left (\frac {4 \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx}{3 d^2}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {4 \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx}{3 d^2}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3043

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3051

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3052

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3389

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3043

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3389

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3050

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3052

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a d}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3385

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {d \sin (e+f x)}}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}-\frac {b \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a d \sqrt {d \sin (e+f x)}}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3384

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {a \left (\frac {8 \sqrt {d \sin (e+f x)}}{3 d^3 f g \sqrt {g \cos (e+f x)}}-\frac {2}{3 d f g (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}\right )-\frac {b \left (\frac {2 \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d^2}-\frac {2}{d f g \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}\right )}{d}}{a^2-b^2}-\frac {b^2 \left (-\frac {b \left (\frac {4 \sqrt {2} b g \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{a d f \sqrt {d \sin (e+f x)}}+\frac {-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}}{a}\right )}{a d}-\frac {2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}\right )}{g^2 \left (a^2-b^2\right )}\)

Input:

Int[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])) 
,x]
 

Output:

-((b^2*((-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[Sqr 
t[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(Sqrt[-a + b]*Sqr 
t[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*C 
os[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^2 - b^2)*g^2)) + (a*(-2/(3*d*f*g*Sqrt[g*Cos[e + 
f*x]]*(d*Sin[e + f*x])^(3/2)) + (8*Sqrt[d*Sin[e + f*x]])/(3*d^3*f*g*Sqrt[g 
*Cos[e + f*x]])) - (b*(-2/(d*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]] 
) + (2*((2*(d*Sin[e + f*x])^(3/2))/(d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[ 
g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(f*g^2* 
Sqrt[Sin[2*e + 2*f*x]])))/d^2))/d)/(a^2 - b^2)
 

Defintions of rubi rules used

rule 993

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]
 

rule 1542

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3043

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]
 

rule 3050

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]
 

rule 3051

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[(-(b*Sin[e + f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1) 
/(a*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1))   Int[(b*Sin[e + f*x 
])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m 
, -1] && IntegersQ[2*m, 2*n]
 

rule 3052

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]
 

rule 3119

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]
 

rule 3317

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a   Int[(g*Co 
s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d   Int[(g*Cos[e + f*x])^ 
p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
 

rule 3383

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 - b^2 
)   Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x], x] 
- Simp[b^2/(g^2*(a^2 - b^2))   Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[e + f*x 
])^n/(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[p, -1]
 

rule 3384

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]
 

rule 3385

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]
 

rule 3389

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1429\) vs. \(2(589)=1178\).

Time = 4.40 (sec) , antiderivative size = 1430, normalized size of antiderivative = 2.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1430\)

Input:

int(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x,method= 
_RETURNVERBOSE)
 

Output:

1/3/f/g/(g*cos(f*x+e))^(1/2)/d^2/(d*sin(f*x+e))^(1/2)*((3*cos(f*x+e)+3)*El 
lipticPi((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^4*(-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)+cot(f*x+e))^(1/2)+(3*cos(f*x+e)+3)*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))*a*b^4*(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)+(3*cos(f*x+e)+3)*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*(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)+c 
ot(f*x+e))^(1/2)+(3*cos(f*x+e)+3)*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^4*(-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)+cot 
(f*x+e))^(1/2)+(-3*cos(f*x+e)-3)*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*(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)+( 
-3*cos(f*x+e)-3)*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*(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)+(24*cos(f*x+e)+24)* 
(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)+1)^(1/2),1/2...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, 
algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate(1/(g*cos(f*x+e))**(3/2)/(d*sin(f*x+e))**(5/2)/(a+b*sin(f*x+e)),x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int { \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, 
algorithm="maxima")
 

Output:

integrate(1/((g*cos(f*x + e))^(3/2)*(b*sin(f*x + e) + a)*(d*sin(f*x + e))^ 
(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int { \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, 
algorithm="giac")
 

Output:

integrate(1/((g*cos(f*x + e))^(3/2)*(b*sin(f*x + e) + a)*(d*sin(f*x + e))^ 
(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int \frac {1}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:

int(1/((g*cos(e + f*x))^(3/2)*(d*sin(e + f*x))^(5/2)*(a + b*sin(e + f*x))) 
,x)
 

Output:

int(1/((g*cos(e + f*x))^(3/2)*(d*sin(e + f*x))^(5/2)*(a + b*sin(e + f*x))) 
, x)
 

Reduce [F]

\[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\frac {\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}}{\cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{4} b +\cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{3} a}d x \right )}{d^{3} g^{2}} \] Input:

int(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(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) 
**2*sin(e + f*x)**4*b + cos(e + f*x)**2*sin(e + f*x)**3*a),x))/(d**3*g**2)