\(\int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx\) [724]

Optimal result

Integrand size = 23, antiderivative size = 455 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))} \] Output:

1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/ 
2*c),2^(1/2))*sec(d*x+c)^(1/2)/a^4/(a^2-b^2)^2/d+1/12*(8*a^4-61*a^2*b^2+35 
*b^4)*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^( 
1/2)/a^3/(a^2-b^2)^2/d+1/4*b^2*(63*a^4-86*a^2*b^2+35*b^4)*cos(d*x+c)^(1/2) 
*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*sec(d*x+c)^(1/2)/a^4/(a- 
b)^2/(a+b)^3/d-1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*sec(d*x+c)^(1/2)*sin(d*x+c 
)/a^4/(a^2-b^2)^2/d+1/12*(8*a^4-61*a^2*b^2+35*b^4)*sec(d*x+c)^(3/2)*sin(d* 
x+c)/a^3/(a^2-b^2)^2/d+1/2*b^2*sec(d*x+c)^(7/2)*sin(d*x+c)/a/(a^2-b^2)/d/( 
b+a*sec(d*x+c))^2+1/4*b^2*(13*a^2-7*b^2)*sec(d*x+c)^(5/2)*sin(d*x+c)/a^2/( 
a^2-b^2)^2/d/(b+a*sec(d*x+c))
 

Mathematica [A] (warning: unable to verify)

Time = 6.63 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.64 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \left (16 a^6+328 a^4 b^2-641 a^2 b^4+315 b^6\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (160 a^5 b-512 a^3 b^3+280 a b^5\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (72 a^4 b^2-195 a^2 b^4+105 b^6\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2}-\frac {b^3 \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {3 \left (5 a^2 b^3 \sin (c+d x)-3 b^5 \sin (c+d x)\right )}{4 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 \tan (c+d x)}{3 a^3}\right )}{d} \] Input:

Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3,x]
 

Output:

((2*(16*a^6 + 328*a^4*b^2 - 641*a^2*b^4 + 315*b^6)*Cos[c + d*x]^2*(Ellipti 
cF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c 
+ d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x]) 
/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(160*a^5*b - 512*a^3*b 
^3 + 280*a*b^5)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x] 
]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a 
+ b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((72*a^4*b^2 - 195*a^2*b^4 + 105 
*b^6)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 
 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt 
[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], 
 -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b) 
, ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x] 
^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c 
+ d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x] 
)*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^4*( 
a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*(b*(24*a^4 - 65*a^2*b^2 
+ 35*b^4)*Sin[c + d*x])/(a^4*(a^2 - b^2)^2) - (b^3*Sin[c + d*x])/(2*a^2*(a 
^2 - b^2)*(a + b*Cos[c + d*x])^2) - (3*(5*a^2*b^3*Sin[c + d*x] - 3*b^5*Sin 
[c + d*x]))/(4*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/ 
(3*a^3)))/d
 

Rubi [A] (verified)

Time = 3.54 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.98, number of steps used = 26, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.130, Rules used = {3042, 3717, 3042, 4332, 27, 3042, 4586, 27, 3042, 4590, 27, 3042, 4590, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}

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.

\(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 3717

\(\displaystyle \int \frac {\sec ^{\frac {11}{2}}(c+d x)}{(a \sec (c+d x)+b)^3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\)

\(\Big \downarrow \) 4332

\(\displaystyle \frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (5 b^2-4 a \sec (c+d x) b+\left (4 a^2-7 b^2\right ) \sec ^2(c+d x)\right )}{2 (b+a \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (5 b^2-4 a \sec (c+d x) b+\left (4 a^2-7 b^2\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (5 b^2-4 a \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (4 a^2-7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4586

\(\displaystyle \frac {\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (3 \left (13 a^2-7 b^2\right ) b^2-4 a \left (4 a^2-b^2\right ) \sec (c+d x) b+\left (8 a^4-61 b^2 a^2+35 b^4\right ) \sec ^2(c+d x)\right )}{2 (b+a \sec (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (3 \left (13 a^2-7 b^2\right ) b^2-4 a \left (4 a^2-b^2\right ) \sec (c+d x) b+\left (8 a^4-61 b^2 a^2+35 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (3 \left (13 a^2-7 b^2\right ) b^2-4 a \left (4 a^2-b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (8 a^4-61 b^2 a^2+35 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4590

\(\displaystyle \frac {\frac {\frac {2 \int \frac {\sqrt {\sec (c+d x)} \left (-3 b \left (24 a^4-65 b^2 a^2+35 b^4\right ) \sec ^2(c+d x)+4 a \left (2 a^4+14 b^2 a^2-7 b^4\right ) \sec (c+d x)+b \left (8 a^4-61 b^2 a^2+35 b^4\right )\right )}{2 (b+a \sec (c+d x))}dx}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (-3 b \left (24 a^4-65 b^2 a^2+35 b^4\right ) \sec ^2(c+d x)+4 a \left (2 a^4+14 b^2 a^2-7 b^4\right ) \sec (c+d x)+b \left (8 a^4-61 b^2 a^2+35 b^4\right )\right )}{b+a \sec (c+d x)}dx}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (-3 b \left (24 a^4-65 b^2 a^2+35 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 a \left (2 a^4+14 b^2 a^2-7 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (8 a^4-61 b^2 a^2+35 b^4\right )\right )}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4590

\(\displaystyle \frac {\frac {\frac {\frac {2 \int \frac {3 \left (24 a^4-65 b^2 a^2+35 b^4\right ) b^2+4 a \left (20 a^4-64 b^2 a^2+35 b^4\right ) \sec (c+d x) b+\left (8 a^6+128 b^2 a^4-223 b^4 a^2+105 b^6\right ) \sec ^2(c+d x)}{2 \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (24 a^4-65 b^2 a^2+35 b^4\right ) b^2+4 a \left (20 a^4-64 b^2 a^2+35 b^4\right ) \sec (c+d x) b+\left (8 a^6+128 b^2 a^4-223 b^4 a^2+105 b^6\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (24 a^4-65 b^2 a^2+35 b^4\right ) b^2+4 a \left (20 a^4-64 b^2 a^2+35 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (8 a^6+128 b^2 a^4-223 b^4 a^2+105 b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4594

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)}dx+\frac {\int \frac {3 \left (24 a^4-65 b^2 a^2+35 b^4\right ) b^3+a \left (8 a^4-61 b^2 a^2+35 b^4\right ) \sec (c+d x) b^2}{\sqrt {\sec (c+d x)}}dx}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {3 \left (24 a^4-65 b^2 a^2+35 b^4\right ) b^3+a \left (8 a^4-61 b^2 a^2+35 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \int \sqrt {\sec (c+d x)}dx+3 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+3 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 4336

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx+\frac {\frac {2 a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {2 a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac {\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}+\frac {\frac {\frac {6 b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}+\frac {\frac {2 a b^2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

Input:

Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3,x]
 

Output:

(b^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d* 
x])^2) + ((b^2*(13*a^2 - 7*b^2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - 
 b^2)*d*(b + a*Sec[c + d*x])) + ((2*(8*a^4 - 61*a^2*b^2 + 35*b^4)*Sec[c + 
d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((((6*b^3*(24*a^4 - 65*a^2*b^2 + 35*b^4 
)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2* 
a*b^2*(8*a^4 - 61*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x) 
/2, 2]*Sqrt[Sec[c + d*x]])/d)/b^2 + (6*b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)* 
Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + 
d*x]])/((a + b)*d))/a - (6*b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sqrt[Sec[c + d 
*x]]*Sin[c + d*x])/(a*d))/(3*a))/(2*a*(a^2 - b^2)))/(4*a*(a^2 - b^2))
 

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 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 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && GtQ[c + d, 0]
 

rule 3717

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Csc[e + f*x])^(m - n*p 
)*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4332

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_), x_Symbol] :> Simp[(-a^2)*d^3*Cot[e + f*x]*(a + b*Csc[e + f*x])^( 
m + 1)*((d*Csc[e + f*x])^(n - 3)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[d^3/ 
(b*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x]) 
^(n - 3)*Simp[a^2*(n - 3) + a*b*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*( 
m + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 
- b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || (IntegersQ[n + 1/2, 2*m] && GtQ[n 
, 2]))
 

rule 4336

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

rule 4586

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + 
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) 
), x] + Simp[d/(b*(a^2 - b^2)*(m + 1))   Int[(a + b*Csc[e + f*x])^(m + 1)*( 
d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A 
 - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n 
+ b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C 
}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
 

rule 4590

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 
)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) 
   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( 
A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc 
[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 
 - b^2, 0] && GtQ[n, 0]
 

rule 4594

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2)   Int[(d*Csc[e + 
f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2   Int[(a*A - (A*b - a 
*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, 
B, C}, x] && NeQ[a^2 - b^2, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2100\) vs. \(2(426)=852\).

Time = 290.12 (sec) , antiderivative size = 2101, normalized size of antiderivative = 4.62

method	result	size
default	\(\text {Expression too large to display}\)	\(2101\)

Input:

int(sec(d*x+c)^(5/2)/(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/a^3*(-1/6*co 
s(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos 
(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+ 
1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ell 
ipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*b^2/a^2*(-1/2*b^2/a/(a^2-b^2)*cos(1/ 
2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*b*cos 
(1/2*d*x+1/2*c)^2+a-b)^2-3/4*b^2*(3*a^2-b^2)/a^2/(a^2-b^2)^2*cos(1/2*d*x+1 
/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*b*cos(1/2*d* 
x+1/2*c)^2+a-b)-7/8/(a+b)/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1 
/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1 
/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/4/(a+b)/(a^2-b^2)/a*(sin(1/2*d 
*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c 
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b+3/ 
8/(a+b)/(a^2-b^2)/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^ 
2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF( 
cos(1/2*d*x+1/2*c),2^(1/2))*b^2-9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^( 
1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d* 
x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+3/8*b^3/a^2/(a^2-b 
^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*s 
in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**(5/2)/(a+b*cos(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")
 

Output:

integrate(sec(d*x + c)^(5/2)/(b*cos(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

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

int((1/cos(c + d*x))^(5/2)/(a + b*cos(c + d*x))^3,x)
 

Output:

int((1/cos(c + d*x))^(5/2)/(a + b*cos(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:

int(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x)
 

Output:

int((sqrt(sec(c + d*x))*sec(c + d*x)**2)/(cos(c + d*x)**3*b**3 + 3*cos(c + 
 d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)