\(\int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [206]

Optimal result

Integrand size = 23, antiderivative size = 456 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {74461 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10 a^5 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^5}-\frac {7 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{32 a^5 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^4}-\frac {155 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{384 a^5 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^3}-\frac {2473 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^2}-\frac {9467 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{4096 a^5 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(5/2)/d+74461/65536 
*2^(1/2)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))/a^( 
5/2)/d+8925/32768*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^3/d-41693/49152*cot( 
d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^4/d+58077/40960*cot(d*x+c)^5*(a+a*sec(d* 
x+c))^(5/2)/a^5/d-1/10*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^5/d/(2+tan(d* 
x+c)^2/(1+sec(d*x+c)))^5-7/32*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^5/d/(2 
+tan(d*x+c)^2/(1+sec(d*x+c)))^4-155/384*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2 
)/a^5/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))^3-2473/3072*cot(d*x+c)^5*(a+a*sec( 
d*x+c))^(5/2)/a^5/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))^2-9467/4096*cot(d*x+c) 
^5*(a+a*sec(d*x+c))^(5/2)/a^5/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))
 

Mathematica [A] (warning: unable to verify)

Time = 4.88 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.64 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \left (-\frac {\sqrt {\frac {1}{2+2 \cos (c+d x)}} (3364685+2115266 \cos (c+d x)+3550428 \cos (2 (c+d x))+1005782 \cos (3 (c+d x))+714844 \cos (4 (c+d x))-1338430 \cos (5 (c+d x))+1168164 \cos (6 (c+d x))+1363110 \cos (7 (c+d x))+639063 \cos (8 (c+d x))) \csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)}}{8192}+1116915 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}-983040 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right )}{122880 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} (a (1+\sec (c+d x)))^{5/2}} \] Input:

Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]
 

Output:

(Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2)*(-1/8192*(Sqrt[(2 + 2*Cos[c + d*x]) 
^(-1)]*(3364685 + 2115266*Cos[c + d*x] + 3550428*Cos[2*(c + d*x)] + 100578 
2*Cos[3*(c + d*x)] + 714844*Cos[4*(c + d*x)] - 1338430*Cos[5*(c + d*x)] + 
1168164*Cos[6*(c + d*x)] + 1363110*Cos[7*(c + d*x)] + 639063*Cos[8*(c + d* 
x)])*Csc[(c + d*x)/2]^5*Sec[(c + d*x)/2]^9*Sqrt[Sec[c + d*x]]) + 1116915*A 
rcSin[Tan[(c + d*x)/2]]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x 
]] - 983040*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^(-1)]] 
*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x]]))/(122880*d*Sqrt[Sec 
[(c + d*x)/2]^2]*(a*(1 + Sec[c + d*x]))^(5/2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.02, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4375, 374, 27, 441, 25, 27, 441, 27, 441, 27, 441, 27, 445, 27, 445, 27, 445, 25, 27, 397, 216}

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.

\(\displaystyle \int \frac {\cot ^6(c+d x)}{(a \sec (c+d x)+a)^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\)

\(\Big \downarrow \) 4375

\(\displaystyle -\frac {2 \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^6}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{a^5 d}\)

\(\Big \downarrow \) 374

\(\displaystyle -\frac {2 \left (\frac {\int \frac {5 a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (1-\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^5}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{20 a}+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (1-\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^5}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {\int -\frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {91 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+27\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^4}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{16 a}+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}-\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {91 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+27\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^4}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{16 a}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}-\frac {1}{16} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {91 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+27\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^4}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}-\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {1705 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+937\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{12 a}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}-\frac {1}{12} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {1705 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+937\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {\int \frac {3 a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {7419 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+5371\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {7419 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+5371\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {66269 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+58077\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {66269 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+58077\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{10} \int \frac {5 a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {58077 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+41693\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {58077 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+41693\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{6} \int \frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {41693 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+8925\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {41693 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+8925\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {8925}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int -\frac {a \left (56611-\frac {8925 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {1}{2} \int \frac {a \left (56611-\frac {8925 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {8925}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {1}{2} a \int \frac {56611-\frac {8925 a \tan ^2(c+d x)}{\sec (c+d x) a+a}}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {8925}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (65536 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-74461 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {8925}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {2473 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {58077}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {41693}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {74461 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {65536 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )+\frac {8925}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}\right )\right )\right )-\frac {9467 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {155 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {7 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^5}\right )}{a^5 d}\)

Input:

Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]
 

Output:

(-2*((Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(20*(2 + (a*Tan[c + d*x]^ 
2)/(a + a*Sec[c + d*x]))^5) + ((7*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2 
))/(16*(2 + (a*Tan[c + d*x]^2)/(a + a*Sec[c + d*x]))^4) + ((155*Cot[c + d* 
x]^5*(a + a*Sec[c + d*x])^(5/2))/(12*(2 + (a*Tan[c + d*x]^2)/(a + a*Sec[c 
+ d*x]))^3) + ((2473*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(8*(2 + (a 
*Tan[c + d*x]^2)/(a + a*Sec[c + d*x]))^2) - (3*(((58077*Cot[c + d*x]^5*(a 
+ a*Sec[c + d*x])^(5/2))/10 - (a*((41693*Cot[c + d*x]^3*(a + a*Sec[c + d*x 
])^(3/2))/6 - (a*((a*((-65536*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec 
[c + d*x]]])/Sqrt[a] + (74461*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[ 
a + a*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a])))/2 + (8925*Cot[c + d*x]*Sqrt[a + 
 a*Sec[c + d*x]])/2))/2))/2)/4 - (9467*Cot[c + d*x]^5*(a + a*Sec[c + d*x]) 
^(5/2))/(4*(2 + (a*Tan[c + d*x]^2)/(a + a*Sec[c + d*x])))))/8)/12)/16)/4)) 
/(a^5*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 441

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
+ b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si 
mp[1/(a*2*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 
)^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m 
 + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, 
 x] && LtQ[p, -1]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 3042

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

rule 4375

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n 
_.), x_Symbol] :> Simp[-2*(a^(m/2 + n + 1/2)/d)   Subst[Int[x^m*((2 + a*x^2 
)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x] 
]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && I 
ntegerQ[n - 1/2]
 

Maple [A] (warning: unable to verify)

Time = 1.16 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.34

method	result	size
default	\(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (220200960 \cos \left (d x +c \right )^{6}[phone] \cos \left (d x +c \right )^{5}[phone] \cos \left (d x +c \right )^{4}[phone] \cos \left (d x +c \right )^{3}[phone] \cos \left (d x +c \right )^{2}[phone] \cos \left (d x +c \right )[phone]\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (250188960 \cos \left (d x +c \right )^{6}[phone] \cos \left (d x +c \right )^{5}[phone] \cos \left (d x +c \right )^{4}[phone] \cos \left (d x +c \right )^{3}[phone] \cos \left (d x +c \right )^{2}[phone] \cos \left (d x +c \right )[phone]\right ) \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (-45714801 \cos \left (d x +c \right )^{10}[phone] \cos \left (d x +c \right )^{9}[phone] \cos \left (d x +c \right )^{8}[phone] \cos \left (d x +c \right )^{7}-280163026 \cos \left (d x +c \right )^{6}-1980132528 \cos \left (d x +c \right )^{5}-1509947582 \cos \left (d x +c \right )^{4}[phone] \cos \left (d x +c \right )^{3}[phone] \cos \left (d x +c \right )^{2}[phone] \cos \left (d x +c \right )[phone]\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}+\left (-377729826 \cos \left (d x +c \right )^{10}-590173398 \cos \left (d x +c \right )^{9}[phone] \cos \left (d x +c \right )^{8}[phone] \cos \left (d x +c \right )^{7}[phone] \cos \left (d x +c \right )^{6}-1086946444 \cos \left (d x +c \right )^{5}-1650473072 \cos \left (d x +c \right )^{4}-299244832 \cos \left (d x +c \right )^{3}[phone] \cos \left (d x +c \right )^{2}[phone] \cos \left (d x +c \right )[phone]\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}\right )}{220200960 d \,a^{3} \left (1+\cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )^{5}+5 \cos \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )\right )}\)	\(610\)

Input:

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/220200960/d/a^3*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(1+cos(d*x+c)^5+ 
5*cos(d*x+c)^4+10*cos(d*x+c)^3+10*cos(d*x+c)^2+5*cos(d*x+c))*((220200960*c 
os(d*x+c)^6+1321205760*cos(d*x+c)^5+3303014400*cos(d*x+c)^4+4404019200*cos 
(d*x+c)^3+3303014400*cos(d*x+c)^2+1321205760*cos(d*x+c)[phone])*2^(1/2) 
*arctanh(2^(1/2)*(-csc(d*x+c)+cot(d*x+c))/(cot(d*x+c)^2-2*csc(d*x+c)*cot(d 
*x+c)+csc(d*x+c)^2-1)^(1/2))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+(2501889 
60*cos(d*x+c)^6+1501133760*cos(d*x+c)^5+3752834400*cos(d*x+c)^4+5003779200 
*cos(d*x+c)^3+3752834400*cos(d*x+c)^2+1501133760*cos(d*x+c)[phone])*ln( 
(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-cot(d*x+c)+csc(d*x+c))*(-2*cos(d*x+c) 
/(1+cos(d*x+c)))^(1/2)+(-45714801*cos(d*x+c)^10+241317156*cos(d*x+c)^9+107 
2093509*cos(d*x+c)^8+1266667336*cos(d*x+c)^7-280163026*cos(d*x+c)^6-198013 
2528*cos(d*x+c)^5-1509947582*cos(d*x+c)^4+192321272*cos(d*x+c)^3+858466323 
*cos(d*x+c)^2+455987532*cos(d*x+c)[phone])*2^(1/2)*(-cos(d*x+c)/(1+cos(d 
*x+c)))^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cot(d*x+c)*csc(d*x+c)^4 
+(-377729826*cos(d*x+c)^10-590173398*cos(d*x+c)^9+236978592*cos(d*x+c)^8+1 
695490864*cos(d*x+c)^7+1142658220*cos(d*x+c)^6-1086946444*cos(d*x+c)^5-165 
0473072*cos(d*x+c)^4-299244832*cos(d*x+c)^3+533539846*cos(d*x+c)^2+3359240 
50*cos(d*x+c)[phone])*cot(d*x+c)*csc(d*x+c)^4)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1023, normalized size of antiderivative = 2.24 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

[-1/1966080*(1116915*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x 
+ c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x 
+ c) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d 
*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) 
 - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 983040*(cos(d* 
x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d* 
x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*log(-(8*a*cos(d*x 
 + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c 
) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 
1))*sin(d*x + c) + 4*(639063*cos(d*x + c)^8 + 681555*cos(d*x + c)^7 - 9860 
85*cos(d*x + c)^6 - 1360025*cos(d*x + c)^5 + 405445*cos(d*x + c)^4 + 83692 
1*cos(d*x + c)^3 + 15305*cos(d*x + c)^2 - 133875*cos(d*x + c))*sqrt((a*cos 
(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c 
)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c) 
^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c)), - 
1/983040*(1116915*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c 
)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c 
) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos( 
d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 983040*(cos(d*x + c)^7 + 3 
*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 ...
 

Sympy [F]

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

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**(5/2),x)
 

Output:

Integral(cot(c + d*x)**6/(a*(sec(c + d*x) + 1))**(5/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {{\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {91 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {3043 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {47185 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {349965 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {1024 \, \sqrt {2} {\left (345 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{8} - 1230 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{6} a + 1760 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} a^{2} - 1150 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a^{3} + 299 \, a^{4}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{5} \sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{983040 \, d} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

1/983040*((2*(4*(6*(8*sqrt(2)*tan(1/2*d*x + 1/2*c)^2/(a^3*sgn(cos(d*x + c) 
)) - 91*sqrt(2)/(a^3*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 3043*sqr 
t(2)/(a^3*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 47185*sqrt(2)/(a^3* 
sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 349965*sqrt(2)/(a^3*sgn(cos(d 
*x + c))))*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*tan(1/2*d*x + 1/2*c) - 1024 
*sqrt(2)*(345*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c 
)^2 + a))^8 - 1230*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 
1/2*c)^2 + a))^6*a + 1760*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2 
*d*x + 1/2*c)^2 + a))^4*a^2 - 1150*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(- 
a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^3 + 299*a^4)/(((sqrt(-a)*tan(1/2*d*x + 
1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^5*sqrt(-a)*a*sgn(cos( 
d*x + c))))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(5/2),x)
 

Output:

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(5/2), x)
 

Reduce [F]

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

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x)
 

Output:

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