\(\int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx\) [386]

Optimal result

Integrand size = 25, antiderivative size = 277 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {1015 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}-\frac {629 \sqrt {\sec (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {23 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {109 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {193 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}} \] Output:

1015/128*arctan(1/2*a^(1/2)*sin(d*x+c)*2^(1/2)/cos(d*x+c)^(1/2)/(a+a*cos(d 
*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)*2^(1/2)/a^(7/2)/d-629/64*s 
ec(d*x+c)^(1/2)*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^(1/2)-1/6*sec(d*x+c)^(3/ 
2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(7/2)-23/48*sec(d*x+c)^(3/2)*sin(d*x+c)/a 
/d/(a+a*cos(d*x+c))^(5/2)-109/64*sec(d*x+c)^(3/2)*sin(d*x+c)/a^2/d/(a+a*co 
s(d*x+c))^(3/2)+193/64*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^ 
(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 9.14 (sec) , antiderivative size = 696, normalized size of antiderivative = 2.51 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2),x]
 

Output:

(Cot[c/2 + (d*x)/2]^7*Csc[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^6*((1 - 2*Sin[ 
c/2 + (d*x)/2]^2)^(-1))^(7/2)*(-7680*Cos[(c + d*x)/2]^10*HypergeometricPFQ 
[{2, 2, 2, 2, 2, 7/2}, {1, 1, 1, 1, 15/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Si 
n[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 + 19200*Cos[(c + d*x)/2]^8*Hype 
rgeometricPFQ[{2, 2, 2, 2, 7/2}, {1, 1, 1, 15/2}, Sin[c/2 + (d*x)/2]^2/(-1 
 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14*(-7 + 6*Sin[c/2 + (d*x)/ 
2]^2) + 143*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 
 2*Sin[c/2 + (d*x)/2]^2)]*(315*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*S 
in[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*(351384 - 2928877*Sin[c/2 + (d*x 
)/2]^2 + 9953934*Sin[c/2 + (d*x)/2]^4 - 17629526*Sin[c/2 + (d*x)/2]^6 + 17 
139064*Sin[c/2 + (d*x)/2]^8 - 8670660*Sin[c/2 + (d*x)/2]^10 + 1793816*Sin[ 
c/2 + (d*x)/2]^12) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^ 
2)]*(-110685960 + 1291549455*Sin[c/2 + (d*x)/2]^2 - 6601900452*Sin[c/2 + ( 
d*x)/2]^4 + 19406027859*Sin[c/2 + (d*x)/2]^6 - 36160322412*Sin[c/2 + (d*x) 
/2]^8 + 44313222590*Sin[c/2 + (d*x)/2]^10 - 35736693140*Sin[c/2 + (d*x)/2] 
^12 + 18305254212*Sin[c/2 + (d*x)/2]^14 - 5410719584*Sin[c/2 + (d*x)/2]^16 
 + 704274992*Sin[c/2 + (d*x)/2]^18))))/(3243240*d*(a*(1 + Cos[c + d*x]))^( 
7/2))
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.08, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4710, 3042, 3245, 27, 3042, 3457, 27, 3042, 3457, 27, 3042, 3463, 27, 3042, 3463, 27, 3042, 3261, 218}

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.

\(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a \cos (c+d x)+a)^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4710

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{7/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{7/2}}dx\)

\(\Big \downarrow \) 3245

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {15 a-8 a \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {15 a-8 a \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {15 a-8 a \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3457

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int \frac {3 \left (63 a^2-46 a^2 \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \int \frac {63 a^2-46 a^2 \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \int \frac {63 a^2-46 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3457

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\int \frac {579 a^3-436 a^3 \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\int \frac {579 a^3-436 a^3 \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\int \frac {579 a^3-436 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3463

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {2 \int -\frac {3 \left (629 a^4-386 a^4 \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{3 a}+\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {629 a^4-386 a^4 \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}}dx}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {629 a^4-386 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3463

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2 \int -\frac {1015 a^5}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}+\frac {1258 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {1258 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-1015 a^4 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {1258 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-1015 a^4 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 3261

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {2030 a^5 \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}+\frac {1258 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {3 \left (\frac {\frac {386 a^3 \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}-\frac {\frac {1258 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {1015 \sqrt {2} a^{7/2} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{a}}{4 a^2}-\frac {109 a^2 \sin (c+d x)}{2 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}-\frac {23 a \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}\right )\)

Input:

Int[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/6*Sin[c + d*x]/(d*Cos[c + d*x]^( 
3/2)*(a + a*Cos[c + d*x])^(7/2)) + ((-23*a*Sin[c + d*x])/(4*d*Cos[c + d*x] 
^(3/2)*(a + a*Cos[c + d*x])^(5/2)) + (3*((-109*a^2*Sin[c + d*x])/(2*d*Cos[ 
c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((386*a^3*Sin[c + d*x])/(d*Co 
s[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((-1015*Sqrt[2]*a^(7/2)*ArcTa 
n[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d* 
x]])])/d + (1258*a^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c 
+ d*x]]))/a)/(4*a^2)))/(8*a^2))/(12*a^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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 3042

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

rule 3245

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

rule 3261

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

rule 3457

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

rule 3463

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

rule 4710

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Csc[a 
+ b*x])^m*(c*Sin[a + b*x])^m   Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [A] (verified)

Time = 7.41 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.78

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {a \left (\cos \left (d x +c \right )+1\right )}\, \sec \left (d x +c \right )^{\frac {5}{2}} \left (\cos \left (d x +c \right ) \sin \left (d x +c \right ) \left (1887 \cos \left (d x +c \right )^{4}+5082 \cos \left (d x +c \right )^{3}+4251 \cos \left (d x +c \right )^{2}+896 \cos \left (d x +c \right )-128\right ) \sqrt {2}+\sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (3045 \cos \left (d x +c \right )^{6}+12180 \cos \left (d x +c \right )^{5}+18270 \cos \left (d x +c \right )^{4}+12180 \cos \left (d x +c \right )^{3}+3045 \cos \left (d x +c \right )^{2}\right )\right )}{384 d \left (\cos \left (d x +c \right )+1\right ) \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right ) a^{4}}\)	\(216\)

Input:

int(sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/384/d*2^(1/2)*(a*(cos(d*x+c)+1))^(1/2)*sec(d*x+c)^(5/2)/(cos(d*x+c)+1)/ 
(cos(d*x+c)^3+3*cos(d*x+c)^2+3*cos(d*x+c)+1)*(cos(d*x+c)*sin(d*x+c)*(1887* 
cos(d*x+c)^4+5082*cos(d*x+c)^3+4251*cos(d*x+c)^2+896*cos(d*x+c)-128)*2^(1/ 
2)+(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arcsin(cot(d*x+c)-csc(d*x+c))*(3045*c 
os(d*x+c)^6+12180*cos(d*x+c)^5+18270*cos(d*x+c)^4+12180*cos(d*x+c)^3+3045* 
cos(d*x+c)^2))/a^4
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {3045 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left (1887 \, \cos \left (d x + c\right )^{4} + 5082 \, \cos \left (d x + c\right )^{3} + 4251 \, \cos \left (d x + c\right )^{2} + 896 \, \cos \left (d x + c\right ) - 128\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \] Input:

integrate(sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

-1/384*(3045*sqrt(2)*(cos(d*x + c)^5 + 4*cos(d*x + c)^4 + 6*cos(d*x + c)^3 
 + 4*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*arctan(sqrt(2)*sqrt(a*cos(d*x 
+ c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) + 2*(1887*cos(d*x + c 
)^4 + 5082*cos(d*x + c)^3 + 4251*cos(d*x + c)^2 + 896*cos(d*x + c) - 128)* 
sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^4*d*cos(d*x + 
 c)^5 + 4*a^4*d*cos(d*x + c)^4 + 6*a^4*d*cos(d*x + c)^3 + 4*a^4*d*cos(d*x 
+ c)^2 + a^4*d*cos(d*x + c))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**(5/2)/(a+a*cos(d*x+c))**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

integrate(sec(d*x + c)^(5/2)/(a*cos(d*x + c) + a)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:

int((1/cos(c + d*x))^(5/2)/(a + a*cos(c + d*x))^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int((1/cos(c + d*x))^(5/2)/(a + a*cos(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1}d x \right )}{a^{4}} \] Input:

int(sec(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(7/2),x)
 

Output:

(sqrt(a)*int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1)*sec(c + d*x)**2)/( 
cos(c + d*x)**4 + 4*cos(c + d*x)**3 + 6*cos(c + d*x)**2 + 4*cos(c + d*x) + 
 1),x))/a**4