\(\int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx\) [194]

Optimal result

Integrand size = 23, antiderivative size = 401 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}+\frac {16363 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{8192 \sqrt {2} a^{3/2} d}-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8192 a^2 d}-\frac {8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{8 a^4 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^4}-\frac {29 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{96 a^4 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^3}-\frac {511 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{768 a^4 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^2}-\frac {2045 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{1024 a^4 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/d+16363/16384 
*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^( 
3/2)/d-21/8192*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^2/d-8171/12288*cot(d*x+ 
c)^3*(a+a*sec(d*x+c))^(3/2)/a^3/d+12267/10240*cot(d*x+c)^5*(a+a*sec(d*x+c) 
)^(5/2)/a^4/d-1/8*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^4/d/(2+tan(d*x+c)^ 
2/(1+sec(d*x+c)))^4-29/96*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^4/d/(2+tan 
(d*x+c)^2/(1+sec(d*x+c)))^3-511/768*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^ 
4/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))^2-2045/1024*cot(d*x+c)^5*(a+a*sec(d*x+ 
c))^(5/2)/a^4/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))
 

Mathematica [A] (warning: unable to verify)

Time = 3.85 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\sec ^{\frac {3}{2}}(c+d x) \left (245445 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}-245760 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}-\frac {1}{32} \sqrt {\frac {1}{2+2 \cos (c+d x)}} (355216+831963 \cos (c+d x)-59592 \cos (2 (c+d x))+153657 \cos (3 (c+d x))-207760 \cos (4 (c+d x))+141291 \cos (5 (c+d x))+207048 \cos (6 (c+d x))+151041 \cos (7 (c+d x))) \csc ^6(c+d x) \sqrt {\sec (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{61440 d \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} (a (1+\sec (c+d x)))^{3/2}} \] Input:

Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]
 

Output:

(Sec[c + d*x]^(3/2)*(245445*ArcSin[Tan[(c + d*x)/2]]*Sqrt[(1 + Sec[c + d*x 
])^(-1)]*Sqrt[1 + Sec[c + d*x]] - 245760*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/S 
qrt[(1 + Sec[c + d*x])^(-1)]]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c 
 + d*x]] - (Sqrt[(2 + 2*Cos[c + d*x])^(-1)]*(355216 + 831963*Cos[c + d*x] 
- 59592*Cos[2*(c + d*x)] + 153657*Cos[3*(c + d*x)] - 207760*Cos[4*(c + d*x 
)] + 141291*Cos[5*(c + d*x)] + 207048*Cos[6*(c + d*x)] + 151041*Cos[7*(c + 
 d*x)])*Csc[c + d*x]^6*Sqrt[Sec[c + d*x]]*Tan[(c + d*x)/2])/32))/(61440*d* 
(Sec[(c + d*x)/2]^2)^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4375, 374, 27, 441, 25, 27, 441, 27, 441, 27, 445, 27, 445, 27, 445, 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.

\(\displaystyle \int \frac {\cot ^6(c+d x)}{(a \sec (c+d x)+a)^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\)

\(\Big \downarrow \) 4375

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

\(\Big \downarrow \) 374

\(\displaystyle -\frac {2 \left (\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (3-\frac {13 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^4}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{16 a}+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (3-\frac {13 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^4}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {\int -\frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {319 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+127\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{12 a}+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}-\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {319 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+127\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{12 a}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}-\frac {1}{12} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {319 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+127\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {\int \frac {3 a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {1533 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1021\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {1533 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1021\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {14315 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+12267\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {14315 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+12267\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{10} \int \frac {5 a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {12267 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+8171\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {12267 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+8171\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {8171}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{6} \int -\frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (21-\frac {8171 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (21-\frac {8171 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {8171}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+16405\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {8171}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+16405}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {8171}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (16384 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-16363 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )+\frac {8171}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{16} \left (\frac {1}{12} \left (\frac {511 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {12267}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {16363 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {16384 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )+\frac {8171}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {2045 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {29 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{16 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^4}\right )}{a^4 d}\)

Input:

Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2),x]
 

Output:

(-2*((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) + ((29*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) + ((511*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*(((12267*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/10 - 
(a*((8171*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/6 + (a*(-1/2*(a*((-16 
384*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + (16 
363*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sq 
rt[2]*Sqrt[a]))) + (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/2))/2))/2)/4 
 - (2045*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))/(a^4*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 = 1.01 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.40

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (6881280 \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 )+6881280\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 (-6872460 \cos \left (d x +c \right )^{5}-34362300 \cos \left (d x +c \right )^{4}-68724600 \cos \left (d x +c \right )^{3}-68724600 \cos \left (d x +c \right )^{2}-34362300 \cos \left (d x +c \right )-6872460\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 )}}+\sqrt {2}\, \left (262371 \cos \left (d x +c \right )^{9}-13502607 \cos \left (d x +c \right )^{8}-37435152 \cos \left (d x +c \right )^{7}-22468424 \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}-22575696 \cos \left (d x +c \right )^{2}-17474457 \cos \left (d x +c \right )-3798795\right ) \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 (8983038 \cos \left (d x +c \right )^{9}+3642276 \cos \left (d x +c \right )^{8}-17397372 \cos \left (d x +c \right )^{7}-20368340 \cos \left (d x +c \right )^{6}[phone] \cos \left (d x +c \right )^{5}[phone] \cos \left (d x +c \right )^{4}+2299004 \cos \left (d x +c \right )^{3}-7999068 \cos \left (d x +c \right )^{2}-2933630 \cos \left (d x +c \right )+17640\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}\right )}{6881280 d \,a^{2} \left (1+\cos \left (d x +c \right )\right ) \left (1+\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 )\right )}\)	\(560\)

Input:

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6881280/d/a^2*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(1+cos(d*x+c)^4+4 
*cos(d*x+c)^3+6*cos(d*x+c)^2+4*cos(d*x+c))*((6881280*cos(d*x+c)^5+34406400 
*cos(d*x+c)^4+68812800*cos(d*x+c)^3+68812800*cos(d*x+c)^2+34406400*cos(d*x 
+c)+6881280)*2^(1/2)*arctanh(2^(1/2)/(cot(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c) 
+csc(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c)))*(-2*cos(d*x+c)/(1+cos(d*x+ 
c)))^(1/2)+(-6872460*cos(d*x+c)^5-34362300*cos(d*x+c)^4-68724600*cos(d*x+c 
)^3-68724600*cos(d*x+c)^2-34362300*cos(d*x+c)-6872460)*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)+2^(1/2)*(262371*cos(d*x+c)^9-13502607*cos(d*x+c)^8-37435152*cos(d*x 
+c)^7-22468424*cos(d*x+c)^6+35538910*cos(d*x+c)^5+56840498*cos(d*x+c)^4+13 
603304*cos(d*x+c)^3-22575696*cos(d*x+c)^2-17474457*cos(d*x+c)-3798795)*(-c 
os(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+(8983038*cos(d*x+c)^9+3642276*cos(d*x+c)^8-17397372*cos 
(d*x+c)^7-20368340*cos(d*x+c)^6+11015040*cos(d*x+c)^5+22741412*cos(d*x+c)^ 
4+2299004*cos(d*x+c)^3-7999068*cos(d*x+c)^2-2933630*cos(d*x+c)+17640)*cot( 
d*x+c)*csc(d*x+c)^4)
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.38 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

[-1/491520*(245445*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + 
c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*lo 
g((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) + 245760*(cos(d*x + c)^6 + 2*cos(d*x 
+ c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*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*(151041*cos(d*x + c 
)^7 + 103524*cos(d*x + c)^6 - 228999*cos(d*x + c)^5 - 181256*cos(d*x + c)^ 
4 + 97611*cos(d*x + c)^3 + 82340*cos(d*x + c)^2 + 315*cos(d*x + c))*sqrt(( 
a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d* 
x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + 
 c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c)), -1/245760*(245445*sqr 
t(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^ 
3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*co 
s(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x 
 + c) + 245760*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos 
(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(2*sqrt(a 
)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*...
 

Sympy [F]

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

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**(3/2),x)
 

Output:

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {5 \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {65 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1451 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {13503 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {256 \, \sqrt {2} {\left (555 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{8} - 1950 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{6} a + 2780 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} a^{2} - 1810 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a^{3} + 473 \, a^{4}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{5} \sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{245760 \, d} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

-1/245760*(5*(2*(4*(6*sqrt(2)*tan(1/2*d*x + 1/2*c)^2/(a^2*sgn(cos(d*x + c) 
)) - 65*sqrt(2)/(a^2*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 1451*sqr 
t(2)/(a^2*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 13503*sqrt(2)/(a^2* 
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) + 256*sqrt(2)*(555*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x 
 + 1/2*c)^2 + a))^8 - 1950*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/ 
2*d*x + 1/2*c)^2 + a))^6*a + 2780*(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 - 1810*(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 + 473*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)*sg 
n(cos(d*x + c))))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(3/2),x)
 

Output:

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(3/2), x)
 

Reduce [F]

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

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

Output:

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