\(\int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx\) [1272]

Optimal result

Integrand size = 35, antiderivative size = 712 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^7 (a-b)^2 (a+b)^{5/2} \sqrt {d} f}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^6 (a-b)^2 (a+b)^{5/2} \sqrt {d} f} \] Output:

2/11*cos(f*x+e)^5*(d*sin(f*x+e))^(1/2)/a/d/f/(a+b*sin(f*x+e))^(11/2)-20/99 
*(a^2-b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^2/b^2/d/f/(a+b*sin(f*x+e))^(9 
/2)+80/693*(3*a^2+2*b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^3/b^2/d/f/(a+b* 
sin(f*x+e))^(7/2)-4/231*(5*a^4-17*a^2*b^2+16*b^4)*cos(f*x+e)*(d*sin(f*x+e) 
)^(1/2)/a^4/b^2/(a^2-b^2)/d/f/(a+b*sin(f*x+e))^(5/2)-8/693*(5*a^6-22*a^4*b 
^2+65*a^2*b^4-32*b^6)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^5/b^2/(a^2-b^2)^2/ 
d/f/(a+b*sin(f*x+e))^(3/2)+16/693*b*(93*a^4-93*a^2*b^2+32*b^4)*cos(f*x+e)/ 
a^5/(a^2-b^2)^3/f/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)-16/693*b*(93 
*a^4-93*a^2*b^2+32*b^4)*(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^7/(a-b)^2/(a+b)^(5/2)/d^(1/ 
2)/f-16/693*(45*a^4-48*a^3*b-69*a^2*b^2+24*a*b^3+32*b^4)*(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^6/(a-b)^2/(a+b)^(5/2)/d^(1/2)/f
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.62 (sec) , antiderivative size = 1906, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2) 
),x]
 

Output:

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((2*(a^4*Cos[e + f*x] - 2*a^2*b^2*C 
os[e + f*x] + b^4*Cos[e + f*x]))/(11*a*b^4*(a + b*Sin[e + f*x])^6) - (4*(1 
8*a^4*Cos[e + f*x] - 13*a^2*b^2*Cos[e + f*x] - 5*b^4*Cos[e + f*x]))/(99*a^ 
2*b^4*(a + b*Sin[e + f*x])^5) + (4*(189*a^4*Cos[e + f*x] - 3*a^2*b^2*Cos[e 
 + f*x] + 40*b^4*Cos[e + f*x]))/(693*a^3*b^4*(a + b*Sin[e + f*x])^4) - (4* 
(42*a^6*Cos[e + f*x] - 37*a^4*b^2*Cos[e + f*x] - 17*a^2*b^4*Cos[e + f*x] + 
 16*b^6*Cos[e + f*x]))/(231*a^4*b^4*(a^2 - b^2)*(a + b*Sin[e + f*x])^3) + 
(2*(63*a^8*Cos[e + f*x] - 146*a^6*b^2*Cos[e + f*x] + 151*a^4*b^4*Cos[e + f 
*x] - 260*a^2*b^6*Cos[e + f*x] + 128*b^8*Cos[e + f*x]))/(693*a^5*b^4*(a^2 
- b^2)^2*(a + b*Sin[e + f*x])^2) - (16*(93*a^4*b^2*Cos[e + f*x] - 93*a^2*b 
^4*Cos[e + f*x] + 32*b^6*Cos[e + f*x]))/(693*a^6*(a^2 - b^2)^3*(a + b*Sin[ 
e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + (8*Sqrt[Sin[e + f*x]]*((4*a*(45*a^ 
6 - 114*a^4*b^2 + 101*a^2*b^4 - 32*b^6)*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*Sin[e + f*x])/a)]*S 
qrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Si 
n[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-93*a^5*b + 93*a^3*b^3 - 32*a 
*b^5)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcS 
in[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], ...
 

Rubi [A] (verified)

Time = 3.84 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 3366, 3042, 3370, 27, 3042, 3534, 3042, 3534, 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.

\(\displaystyle \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (e+f x)^6}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}}dx\)

\(\Big \downarrow \) 3366

\(\displaystyle \frac {10 \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{11/2}}dx}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10 \int \frac {\cos (e+f x)^4}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{11/2}}dx}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3370

\(\displaystyle \frac {10 \left (-\frac {4 \int \frac {5 a^2-14 b \sin (e+f x) a-48 b^2-\left (15 a^2-32 b^2\right ) \sin ^2(e+f x)}{4 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {10 \left (-\frac {\int \frac {5 a^2-14 b \sin (e+f x) a-48 b^2-\left (15 a^2-32 b^2\right ) \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10 \left (-\frac {\int \frac {5 a^2-14 b \sin (e+f x) a-48 b^2-\left (15 a^2-32 b^2\right ) \sin (e+f x)^2}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \int \frac {-3 \left (5 a^4-17 b^2 a^2+16 b^4\right ) d \sin ^2(e+f x)-10 a b \left (a^2-4 b^2\right ) d \sin (e+f x)+\left (5 a^4-107 b^2 a^2+96 b^4\right ) d}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}dx}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \int \frac {-3 \left (5 a^4-17 b^2 a^2+16 b^4\right ) d \sin (e+f x)^2-10 a b \left (a^2-4 b^2\right ) d \sin (e+f x)+\left (5 a^4-107 b^2 a^2+96 b^4\right ) d}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}dx}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \int -\frac {3 \left (b^2 \left (45 a^4-69 b^2 a^2+32 b^4\right ) d^2-24 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)\right )}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \int \frac {b^2 \left (45 a^4-69 b^2 a^2+32 b^4\right ) d^2-24 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \int \frac {b^2 \left (45 a^4-69 b^2 a^2+32 b^4\right ) d^2-24 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3472

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \int \frac {\left (93 a^4-93 b^2 a^2+32 b^4\right ) d^2 b^3+a \left (45 a^4-21 b^2 a^2+8 b^4\right ) d^2 \sin (e+f x) b^2}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \int \frac {\left (93 a^4-93 b^2 a^2+32 b^4\right ) d^2 b^3+a \left (45 a^4-21 b^2 a^2+8 b^4\right ) d^2 \sin (e+f x) b^2}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \left (b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx+b^2 d (a-b) \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx\right )}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \left (b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx+b^2 d (a-b) \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx\right )}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3295

\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \left (b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx-\frac {2 b^2 \sqrt {d} (a-b) \sqrt {a+b} \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a f}\right )}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3473

\(\displaystyle \frac {10 \left (\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac {\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {d \left (-\frac {2 b^3 \sqrt {d} (a-b) \sqrt {a+b} \left (93 a^4-93 a^2 b^2+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 f}-\frac {2 b^2 \sqrt {d} (a-b) \sqrt {a+b} \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a f}\right )}{a^2-b^2}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}}{63 a^2 b^2}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\)

Input:

Int[Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2)),x]
 

Output:

(2*Cos[e + f*x]^5*Sqrt[d*Sin[e + f*x]])/(11*a*d*f*(a + b*Sin[e + f*x])^(11 
/2)) + (10*((-2*(a^2 - b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(9*a*b^2*d* 
f*(a + b*Sin[e + f*x])^(9/2)) + (8*(3*a^2 + 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin 
[e + f*x]])/(63*a^2*b^2*d*f*(a + b*Sin[e + f*x])^(7/2)) - ((6*(5*a^4 - 17* 
a^2*b^2 + 16*b^4)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(5*a*(a^2 - b^2)*d*f* 
(a + b*Sin[e + f*x])^(5/2)) + (2*((2*(5*a^6 - 22*a^4*b^2 + 65*a^2*b^4 - 32 
*b^6)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(a*(a^2 - b^2)*f*(a + b*Sin[e + f 
*x])^(3/2)) - (2*((2*b^3*(93*a^4 - 93*a^2*b^2 + 32*b^4)*d^2*Cos[e + f*x])/ 
((a^2 - b^2)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + (d*((-2*(a 
 - b)*b^3*Sqrt[a + b]*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Sqrt[d]*Sqrt[(a*(1 - 
Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[Arc 
Sin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])] 
, -((a + b)/(a - b))]*Tan[e + f*x])/(a^2*f) - (2*(a - b)*b^2*Sqrt[a + b]*( 
45*a^4 - 48*a^3*b - 69*a^2*b^2 + 24*a*b^3 + 32*b^4)*Sqrt[d]*Sqrt[(a*(1 - C 
sc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcS 
in[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], 
 -((a + b)/(a - b))]*Tan[e + f*x])/(a*f)))/(a^2 - b^2)))/(a*(a^2 - b^2)*d) 
))/(5*a*(a^2 - b^2)*d))/(63*a^2*b^2)))/(11*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 3370

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + 
f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 
 1))), x] + (Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*S 
in[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2) 
)), x] - Simp[1/(a^2*b^2*(m + 1)*(m + 2))   Int[(a + b*Sin[e + f*x])^(m + 2 
)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) 
 + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n 
 + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a 
^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m 
, -2] || EqQ[m + n + 4, 0])
 

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 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[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[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, 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. \(11693\) vs. \(2(642)=1284\).

Time = 17.24 (sec) , antiderivative size = 11694, normalized size of antiderivative = 16.42

method	result	size
default	\(\text {Expression too large to display}\)	\(11694\)

Input:

int(cos(f*x+e)^6/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(13/2),x,method=_RE 
TURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^6/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(13/2),x, alg 
orithm="fricas")
 

Output:

integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))*cos(f*x + e)^6/(b^7 
*d*cos(f*x + e)^8 - (21*a^2*b^5 + 4*b^7)*d*cos(f*x + e)^6 + (35*a^4*b^3 + 
63*a^2*b^5 + 6*b^7)*d*cos(f*x + e)^4 - (7*a^6*b + 70*a^4*b^3 + 63*a^2*b^5 
+ 4*b^7)*d*cos(f*x + e)^2 + (7*a^6*b + 35*a^4*b^3 + 21*a^2*b^5 + b^7)*d - 
(7*a*b^6*d*cos(f*x + e)^6 - 7*(5*a^3*b^4 + 3*a*b^6)*d*cos(f*x + e)^4 + 7*( 
3*a^5*b^2 + 10*a^3*b^4 + 3*a*b^6)*d*cos(f*x + e)^2 - (a^7 + 21*a^5*b^2 + 3 
5*a^3*b^4 + 7*a*b^6)*d)*sin(f*x + e)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**6/(d*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(cos(f*x+e)^6/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(13/2),x, alg 
orithm="maxima")
 

Output:

integrate(cos(f*x + e)^6/((b*sin(f*x + e) + a)^(13/2)*sqrt(d*sin(f*x + e)) 
), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)^6/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(13/2),x, alg 
orithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^6}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \] Input:

int(cos(e + f*x)^6/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(13/2)),x)
 

Output:

int(cos(e + f*x)^6/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(13/2)), x 
)
 

Reduce [F]

\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \cos \left (f x +e \right )^{6}}{\sin \left (f x +e \right )^{8} b^{7}+7 \sin \left (f x +e \right )^{7} a \,b^{6}+21 \sin \left (f x +e \right )^{6} a^{2} b^{5}+35 \sin \left (f x +e \right )^{5} a^{3} b^{4}+35 \sin \left (f x +e \right )^{4} a^{4} b^{3}+21 \sin \left (f x +e \right )^{3} a^{5} b^{2}+7 \sin \left (f x +e \right )^{2} a^{6} b +\sin \left (f x +e \right ) a^{7}}d x \right )}{d} \] Input:

int(cos(f*x+e)^6/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(13/2),x)
 

Output:

(sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(sin(e + f*x)*b + a)*cos(e + f*x)**6) 
/(sin(e + f*x)**8*b**7 + 7*sin(e + f*x)**7*a*b**6 + 21*sin(e + f*x)**6*a** 
2*b**5 + 35*sin(e + f*x)**5*a**3*b**4 + 35*sin(e + f*x)**4*a**4*b**3 + 21* 
sin(e + f*x)**3*a**5*b**2 + 7*sin(e + f*x)**2*a**6*b + sin(e + f*x)*a**7), 
x))/d