\(\int \frac {\cot ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx\) [182]

Optimal result

Integrand size = 23, antiderivative size = 346 \[ \int \frac {\cot ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {835 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {189 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{512 a d}-\frac {323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac {579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{6 a^3 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^3}-\frac {23 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{48 a^3 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )^2}-\frac {101 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{64 a^3 d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(1/2)/d+835/1024*ar 
ctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/a^(1/2 
)/d-189/512*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a/d-323/768*cot(d*x+c)^3*(a+ 
a*sec(d*x+c))^(3/2)/a^2/d+579/640*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^3/ 
d-1/6*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^3/d/(2+tan(d*x+c)^2/(1+sec(d*x 
+c)))^3-23/48*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^3/d/(2+tan(d*x+c)^2/(1 
+sec(d*x+c)))^2-101/64*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^3/d/(2+tan(d* 
x+c)^2/(1+sec(d*x+c)))
 

Mathematica [A] (warning: unable to verify)

Time = 3.00 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (-\frac {\left (\frac {1}{1+\cos (c+d x)}\right )^{3/2} (21682+11948 \cos (c+d x)+12791 \cos (2 (c+d x))-14754 \cos (3 (c+d x))+846 \cos (4 (c+d x))+6902 \cos (5 (c+d x))+9737 \cos (6 (c+d x))) \csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)}}{256 \sqrt {2}}+12525 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}-15360 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right )}{7680 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {a (1+\sec (c+d x))}} \] Input:

Integrate[Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]
 

Output:

(Sqrt[Sec[c + d*x]]*(-1/256*(((1 + Cos[c + d*x])^(-1))^(3/2)*(21682 + 1194 
8*Cos[c + d*x] + 12791*Cos[2*(c + d*x)] - 14754*Cos[3*(c + d*x)] + 846*Cos 
[4*(c + d*x)] + 6902*Cos[5*(c + d*x)] + 9737*Cos[6*(c + d*x)])*Csc[(c + d* 
x)/2]^5*Sec[(c + d*x)/2]^3*Sqrt[Sec[c + d*x]])/Sqrt[2] + 12525*ArcSin[Tan[ 
(c + d*x)/2]]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x]] - 15360 
*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]]))/(7680*d*Sqrt[Sec[(c + d*x)/2] 
^2]*Sqrt[a*(1 + Sec[c + d*x])])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 4375, 374, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4375

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

\(\Big \downarrow \) 374

\(\displaystyle -\frac {2 \left (\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (1-\frac {11 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{12 a}+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{12} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (1-\frac {11 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^3}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {\int -\frac {3 a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {69 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+37\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{8 a}+\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {69 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+37\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )^2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {\int \frac {a \cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {707 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+579\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{4 a}-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \int \frac {\cot ^6(c+d x) (\sec (c+d x) a+a)^3 \left (\frac {707 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+579\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{10} \int \frac {5 a \cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {579 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+323\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \int \frac {\cot ^4(c+d x) (\sec (c+d x) a+a)^2 \left (\frac {579 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+323\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {323}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{6} \int -\frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (189-\frac {323 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (189-\frac {323 a \tan ^2(c+d x)}{\sec (c+d x) a+a}\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )+\frac {323}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {189}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {189 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1213\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {323}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {189}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {189 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1213}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )+\frac {323}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {189}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (1024 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-835 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )+\frac {323}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{12} \left (\frac {23 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{8 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {579}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} a \left (\frac {189}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {835 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {1024 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )+\frac {323}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}\right )\right )-\frac {101 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{12 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )^3}\right )}{a^3 d}\)

Input:

Int[Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]
 

Output:

(-2*((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) + ((23*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*(((579*Cot[c 
+ d*x]^5*(a + a*Sec[c + d*x])^(5/2))/10 - (a*((323*Cot[c + d*x]^3*(a + a*S 
ec[c + d*x])^(3/2))/6 + (a*(-1/2*(a*((-1024*ArcTan[(Sqrt[a]*Tan[c + d*x])/ 
Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + (835*ArcTan[(Sqrt[a]*Tan[c + d*x])/(S 
qrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]))) + (189*Cot[c + d*x] 
*Sqrt[a + a*Sec[c + d*x]])/2))/2))/2)/4 - (101*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) 
)/(a^3*d)
 

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 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 = 0.69 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (-245760 \cos \left (d x +c \right )^{4}-983040 \cos \left (d x +c \right )^{3}-1474560 \cos \left (d x +c \right )^{2}-983040 \cos \left (d x +c \right )-245760\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 (200400 \cos \left (d x +c \right )^{4}+801600 \cos \left (d x +c \right )^{3}+1202400 \cos \left (d x +c \right )^{2}+801600 \cos \left (d x +c \right )+200400\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 (59437 \cos \left (d x +c \right )^{8}+711954 \cos \left (d x +c \right )^{7}+1249022 \cos \left (d x +c \right )^{6}-61750 \cos \left (d x +c \right )^{5}-1764360 \cos \left (d x +c \right )^{4}-1156298 \cos \left (d x +c \right )^{3}+457314 \cos \left (d x +c \right )^{2}+702702 \cos \left (d x +c \right )+195195\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {\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 (-192710 \cos \left (d x +c \right )^{8}+259850 \cos \left (d x +c \right )^{7}+387570 \cos \left (d x +c \right )^{6}-380510 \cos \left (d x +c \right )^{5}-543730 \cos \left (d x +c \right )^{4}+278430 \cos \left (d x +c \right )^{3}+357670 \cos \left (d x +c \right )^{2}-75850 \cos \left (d x +c \right )-90720\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}\right )}{245760 d a \left (1+\cos \left (d x +c \right )\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 )}\)	\(510\)

Input:

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/245760/d/a*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(cos(d*x+c)^3+3*cos(d 
*x+c)^2+3*cos(d*x+c)+1)*((-245760*cos(d*x+c)^4-983040*cos(d*x+c)^3-1474560 
*cos(d*x+c)^2-983040*cos(d*x+c)-245760)*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)+(200400*cos(d*x+c)^4+801600*cos(d*x+ 
c)^3+1202400*cos(d*x+c)^2+801600*cos(d*x+c)+200400)*ln((-2*cos(d*x+c)/(1+c 
os(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)*(59437*cos(d*x+c)^8+711954*cos(d*x+c)^7+1249022*cos(d*x+c)^6-6 
1750*cos(d*x+c)^5-1764360*cos(d*x+c)^4-1156298*cos(d*x+c)^3+457314*cos(d*x 
+c)^2+702702*cos(d*x+c)+195195)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-cos 
(d*x+c)/(1+cos(d*x+c)))^(1/2)*cot(d*x+c)*csc(d*x+c)^4+(-192710*cos(d*x+c)^ 
8+259850*cos(d*x+c)^7+387570*cos(d*x+c)^6-380510*cos(d*x+c)^5-543730*cos(d 
*x+c)^4+278430*cos(d*x+c)^3+357670*cos(d*x+c)^2-75850*cos(d*x+c)-90720)*co 
t(d*x+c)*csc(d*x+c)^4)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 823, normalized size of antiderivative = 2.38 \[ \int \frac {\cot ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

[-1/30720*(12525*sqrt(2)*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c) 
^3 - 2*cos(d*x + c)^2 + 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*c 
os(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) + 15360*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^ 
3 - 2*cos(d*x + c)^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*(9737*cos(d*x + c)^6 + 3451*cos(d*x + c)^5 - 14394*cos(d*x + 
 c)^4 - 6158*cos(d*x + c)^3 + 6065*cos(d*x + c)^2 + 2835*cos(d*x + c))*sqr 
t((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + 
 c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a 
*d)*sin(d*x + c)), -1/15360*(12525*sqrt(2)*(cos(d*x + c)^5 + cos(d*x + c)^ 
4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + 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) + 15360*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos( 
d*x + c)^3 - 2*cos(d*x + c)^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*a*co 
s(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(9737*cos(d*x + c)^6 
+ 3451*cos(d*x + c)^5 - 14394*cos(d*x + c)^4 - 6158*cos(d*x + c)^3 + 60...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(cot(d*x + c)^6/sqrt(a*sec(d*x + c) + a), x)
 

Giac [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.45 \[ \int \frac {\cot ^6(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

1/30720*sqrt(2)*(10*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*(2*(4*tan(1/2*d*x 
+ 1/2*c)^2/a - 43/a)*tan(1/2*d*x + 1/2*c)^2 + 567/a)*tan(1/2*d*x + 1/2*c) 
+ 15360*sqrt(2)*sqrt(-a)*log(abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(- 
a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - 4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a) 
*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + 4*sqrt(2) 
*abs(a) - 6*a))/abs(a) - 12525*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(- 
a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/sqrt(-a) + 192*(145*(sqrt(-a)*tan(1/2*d* 
x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*sqrt(-a) - 500*(sqrt(- 
a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*sqrt(-a)* 
a + 710*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + 
a))^4*sqrt(-a)*a^2 - 460*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2* 
d*x + 1/2*c)^2 + a))^2*sqrt(-a)*a^3 + 121*sqrt(-a)*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)/(d*sgn(cos( 
d*x + c)))
 

Mupad [F(-1)]

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

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(1/2),x)
 

Output:

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(1/2), x)
 

Reduce [F]

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

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

Output:

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