\(\int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx\) [1186]

Optimal result

Integrand size = 35, antiderivative size = 510 \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {32 b \left (2 a^2-b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{35 a^5 (a-b) (a+b)^{3/2} \sqrt {d} f}-\frac {8 \left (5 a^2-3 a b-4 b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{35 a^4 (a-b) (a+b)^{3/2} \sqrt {d} f} \] Output:

2/7*cos(f*x+e)^3*(d*sin(f*x+e))^(1/2)/a/d/f/(a+b*sin(f*x+e))^(7/2)+12/35*c 
os(f*x+e)*(d*sin(f*x+e))^(1/2)/a^2/d/f/(a+b*sin(f*x+e))^(5/2)+8/35*(a^2-2* 
b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^3/(a^2-b^2)/d/f/(a+b*sin(f*x+e))^(3 
/2)+32/35*b*(2*a^2-b^2)*cos(f*x+e)/a^3/(a^2-b^2)^2/f/(d*sin(f*x+e))^(1/2)/ 
(a+b*sin(f*x+e))^(1/2)-32/35*b*(2*a^2-b^2)*(a*(1-csc(f*x+e))/(a+b))^(1/2)* 
(a*(1+csc(f*x+e))/(a-b))^(1/2)*EllipticE(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a 
+b)^(1/2)/(d*sin(f*x+e))^(1/2),(-(a+b)/(a-b))^(1/2))*tan(f*x+e)/a^5/(a-b)/ 
(a+b)^(3/2)/d^(1/2)/f-8/35*(5*a^2-3*a*b-4*b^2)*(a*(1-csc(f*x+e))/(a+b))^(1 
/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*EllipticF(d^(1/2)*(a+b*sin(f*x+e))^(1/2 
)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),(-(a+b)/(a-b))^(1/2))*tan(f*x+e)/a^4/(a 
-b)/(a+b)^(3/2)/d^(1/2)/f
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.92 (sec) , antiderivative size = 1670, normalized size of antiderivative = 3.27 \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)) 
,x]
 

Output:

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((-2*(a^2*Cos[e + f*x] - b^2*Cos[e 
+ f*x]))/(7*a*b^2*(a + b*Sin[e + f*x])^4) + (4*(5*a^2*Cos[e + f*x] + 3*b^2 
*Cos[e + f*x]))/(35*a^2*b^2*(a + b*Sin[e + f*x])^3) - (2*(5*a^4*Cos[e + f* 
x] - 9*a^2*b^2*Cos[e + f*x] + 8*b^4*Cos[e + f*x]))/(35*a^3*b^2*(a^2 - b^2) 
*(a + b*Sin[e + f*x])^2) - (32*(2*a^2*b^2*Cos[e + f*x] - b^4*Cos[e + f*x]) 
)/(35*a^4*(a^2 - b^2)^2*(a + b*Sin[e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + 
 (4*Sqrt[Sin[e + f*x]]*((4*a*(5*a^4 - 9*a^2*b^2 + 4*b^4)*Sqrt[((a + b)*Cot 
[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - 
 f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x 
]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Si 
n[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]) 
/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-8*a^3*b + 4 
*a*b^3)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[Ar 
cSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], ( 
-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Cs 
c[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^ 
2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + 
 f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[ 
-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/S 
qrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[...
 

Rubi [A] (verified)

Time = 2.86 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {3042, 3366, 3042, 3368, 3042, 3535, 3042, 3535, 27, 3042, 3472, 3042, 3477, 3042, 3295, 3473}

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.

\(\displaystyle \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (e+f x)^4}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}dx\)

\(\Big \downarrow \) 3366

\(\displaystyle \frac {6 \int \frac {\cos ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \int \frac {\cos (e+f x)^2}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3368

\(\displaystyle \frac {6 \int \frac {1-\sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \int \frac {1-\sin (e+f x)^2}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3535

\(\displaystyle \frac {6 \left (\frac {2 \int \frac {2 \left (a^2-b^2\right ) d-\left (a^2-b^2\right ) d \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}dx}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (\frac {2 \int \frac {2 \left (a^2-b^2\right ) d-\left (a^2-b^2\right ) d \sin (e+f x)^2}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}dx}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3535

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {2 \int \frac {\left (5 a^4-9 b^2 a^2+4 b^4\right ) d^2-3 a b \left (a^2-b^2\right ) d^2 \sin (e+f x)}{2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\int \frac {\left (5 a^4-9 b^2 a^2+4 b^4\right ) d^2-3 a b \left (a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\int \frac {\left (5 a^4-9 b^2 a^2+4 b^4\right ) d^2-3 a b \left (a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3472

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\frac {d \int \frac {4 b \left (2 a^4-3 b^2 a^2+b^4\right ) d^2+a \left (5 a^4-6 b^2 a^2+b^4\right ) \sin (e+f x) d^2}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx}{a^2-b^2}+\frac {8 b d^2 \left (2 a^2-b^2\right ) \cos (e+f x)}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\frac {d \int \frac {4 b \left (2 a^4-3 b^2 a^2+b^4\right ) d^2+a \left (5 a^4-6 b^2 a^2+b^4\right ) \sin (e+f x) d^2}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx}{a^2-b^2}+\frac {8 b d^2 \left (2 a^2-b^2\right ) \cos (e+f x)}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\frac {d \left (d (a+b) \left (5 a^2-3 a b-4 b^2\right ) (a-b)^2 \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx+4 b d^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx\right )}{a^2-b^2}+\frac {8 b d^2 \left (2 a^2-b^2\right ) \cos (e+f x)}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\frac {d \left (d (a+b) \left (5 a^2-3 a b-4 b^2\right ) (a-b)^2 \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx+4 b d^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx\right )}{a^2-b^2}+\frac {8 b d^2 \left (2 a^2-b^2\right ) \cos (e+f x)}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3295

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {\frac {d \left (4 b d^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx-\frac {2 \sqrt {d} (a-b)^2 (a+b)^{3/2} \left (5 a^2-3 a b-4 b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a f}\right )}{a^2-b^2}+\frac {8 b d^2 \left (2 a^2-b^2\right ) \cos (e+f x)}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3473

\(\displaystyle \frac {6 \left (\frac {2 \left (\frac {2 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{3 a f (a+b \sin (e+f x))^{3/2}}+\frac {\frac {8 b d^2 \left (2 a^2-b^2\right ) \cos (e+f x)}{f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {d \left (-\frac {2 \sqrt {d} (a+b)^{3/2} \left (5 a^2-3 a b-4 b^2\right ) (a-b)^2 \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a f}-\frac {8 b \sqrt {d} \sqrt {a+b} \left (2 a^4-3 a^2 b^2+b^4\right ) (a-b) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 f}\right )}{a^2-b^2}}{3 a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {2 \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f (a+b \sin (e+f x))^{5/2}}\right )}{7 a}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}\)

Input:

Int[Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)),x]
 

Output:

(2*Cos[e + f*x]^3*Sqrt[d*Sin[e + f*x]])/(7*a*d*f*(a + b*Sin[e + f*x])^(7/2 
)) + (6*((2*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(5*a*d*f*(a + b*Sin[e + f*x 
])^(5/2)) + (2*((2*(a^2 - 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(3*a*f 
*(a + b*Sin[e + f*x])^(3/2)) + ((8*b*(2*a^2 - b^2)*d^2*Cos[e + f*x])/(f*Sq 
rt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + (d*((-8*(a - b)*b*Sqrt[a + 
b]*(2*a^4 - 3*a^2*b^2 + b^4)*Sqrt[d]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]* 
Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b* 
Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Ta 
n[e + f*x])/(a^2*f) - (2*(a - b)^2*(a + b)^(3/2)*(5*a^2 - 3*a*b - 4*b^2)*S 
qrt[d]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a 
 - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sq 
rt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*f)))/(a^2 - b^2 
))/(3*a*(a^2 - b^2)*d)))/(5*a*(a^2 - b^2)*d)))/(7*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3366

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(-g)*(g* 
Cos[e + f*x])^(p - 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^(m + 1)/(a 
*d*f*(m + 1))), x] + Simp[g^2*((2*m + 3)/(2*a*(m + 1)))   Int[(g*Cos[e + f* 
x])^(p - 2)*((a + b*Sin[e + f*x])^(m + 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; 
FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && EqQ[m + 
p + 1/2, 0]
 

rule 3368

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a 
 + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n 
}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
 

rule 3472

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(d_.)*sin[(e_.) + (f_.)*( 
x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)), x_Symbol] :> Simp[2*(A 
*b - a*B)*(Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[ 
e + f*x]])), x] + Simp[d/(a^2 - b^2)   Int[(A*b - a*B + (a*A - b*B)*Sin[e + 
 f*x])/(Sqrt[a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{ 
a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]
 

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

rule 3535

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S 
in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m 
+ 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin 
[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n 
+ 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d 
*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 
- d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) || 
 !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 
 0])))
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(6927\) vs. \(2(452)=904\).

Time = 9.14 (sec) , antiderivative size = 6928, normalized size of antiderivative = 13.58

method	result	size
default	\(\text {Expression too large to display}\)	\(6928\)

Input:

int(cos(f*x+e)^4/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(9/2),x,method=_RET 
URNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^4/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(9/2),x, algo 
rithm="fricas")
 

Output:

integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))*cos(f*x + e)^4/(b^ 
5*d*cos(f*x + e)^6 - (10*a^2*b^3 + 3*b^5)*d*cos(f*x + e)^4 + (5*a^4*b + 20 
*a^2*b^3 + 3*b^5)*d*cos(f*x + e)^2 - (5*a^4*b + 10*a^2*b^3 + b^5)*d - (5*a 
*b^4*d*cos(f*x + e)^4 - 10*(a^3*b^2 + a*b^4)*d*cos(f*x + e)^2 + (a^5 + 10* 
a^3*b^2 + 5*a*b^4)*d)*sin(f*x + e)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**4/(d*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^4/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(9/2),x, algo 
rithm="maxima")
 

Output:

integrate(cos(f*x + e)^4/((b*sin(f*x + e) + a)^(9/2)*sqrt(d*sin(f*x + e))) 
, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)^4/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(9/2),x, algo 
rithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^4}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \] Input:

int(cos(e + f*x)^4/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(9/2)),x)
 

Output:

int(cos(e + f*x)^4/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(9/2)), x)
 

Reduce [F]

\[ \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \cos \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{6} b^{5}+5 \sin \left (f x +e \right )^{5} a \,b^{4}+10 \sin \left (f x +e \right )^{4} a^{2} b^{3}+10 \sin \left (f x +e \right )^{3} a^{3} b^{2}+5 \sin \left (f x +e \right )^{2} a^{4} b +\sin \left (f x +e \right ) a^{5}}d x \right )}{d} \] Input:

int(cos(f*x+e)^4/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(9/2),x)
 

Output:

(sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(sin(e + f*x)*b + a)*cos(e + f*x)**4) 
/(sin(e + f*x)**6*b**5 + 5*sin(e + f*x)**5*a*b**4 + 10*sin(e + f*x)**4*a** 
2*b**3 + 10*sin(e + f*x)**3*a**3*b**2 + 5*sin(e + f*x)**2*a**4*b + sin(e + 
 f*x)*a**5),x))/d