\(\int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx\) [158]

Optimal result

Integrand size = 23, antiderivative size = 231 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {11 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} d}-\frac {21 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{16 d}+\frac {5 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{24 d}+\frac {3 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{20 a d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{2 a d \left (2+\frac {\tan ^2(c+d x)}{1+\sec (c+d x)}\right )} \] Output:

-2*a^(3/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+11/32*a^(3/ 
2)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))*2^(1/2)/d 
-21/16*a*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d+5/24*cot(d*x+c)^3*(a+a*sec(d* 
x+c))^(3/2)/d+3/20*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a/d-1/2*cot(d*x+c)^ 
5*(a+a*sec(d*x+c))^(5/2)/a/d/(2+tan(d*x+c)^2/(1+sec(d*x+c)))
 

Mathematica [A] (warning: unable to verify)

Time = 5.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {(a (1+\sec (c+d x)))^{3/2} \left (165 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}+2 \sqrt {2} \left (-240 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}+\frac {\sqrt {\frac {1}{1+\cos (c+d x)}} \csc ^5(c+d x) (281-279 \sec (c+d x)+\cos (2 (c+d x)) (-449+351 \sec (c+d x)))}{\sec ^{\frac {3}{2}}(c+d x)}\right )\right )}{960 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x)} \] Input:

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

Output:

((a*(1 + Sec[c + d*x]))^(3/2)*(165*ArcSin[Tan[(c + d*x)/2]]*Sec[(c + d*x)/ 
2]^4*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x]] + 2*Sqrt[2]*(-24 
0*ArcTan[Tan[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^(-1)]]*Sec[(c + d*x)/2]^ 
4*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x]] + (Sqrt[(1 + Cos[c 
+ d*x])^(-1)]*Csc[c + d*x]^5*(281 - 279*Sec[c + d*x] + Cos[2*(c + d*x)]*(- 
449 + 351*Sec[c + d*x])))/Sec[c + d*x]^(3/2))))/(960*d*Sqrt[Sec[(c + d*x)/ 
2]^2]*Sec[c + d*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4375, 374, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4375

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

\(\Big \downarrow \) 374

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {5}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{6} \int \frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {5 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+21\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {3}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )}{a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {5}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {5 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+21\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )-\frac {3}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )}{a d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {5}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+53\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {3}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )}{a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {5}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {21 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+53}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )-\frac {3}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )}{a d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {5}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (32 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-11 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )\right )-\frac {3}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )}{a d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{4} \left (-\frac {1}{2} a \left (\frac {5}{6} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {21}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {11 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {32 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )\right )-\frac {3}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}\right )+\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 \left (\frac {a \tan ^2(c+d x)}{a \sec (c+d x)+a}+2\right )}\right )}{a d}\)

Input:

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

Output:

(-2*(((-3*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/10 - (a*((5*Cot[c + d 
*x]^3*(a + a*Sec[c + d*x])^(3/2))/6 - (a*(-1/2*(a*((-32*ArcTan[(Sqrt[a]*Ta 
n[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + (11*ArcTan[(Sqrt[a]*Tan[c 
 + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]))) + (21*Co 
t[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/2))/2))/2)/4 + (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])))))/ 
(a*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 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 [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs. \(2(198)=396\).

Time = 1.23 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.72

method	result	size
default	\(\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (7680 \cos \left (d x +c \right )^{2}+15360 \cos \left (d x +c \right )+7680\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+\left (2640 \cos \left (d x +c \right )^{2}+5280 \cos \left (d x +c \right )+2640\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\sqrt {2}\, \left (9289 \cos \left (d x +c \right )^{6}+30040 \cos \left (d x +c \right )^{5}+6125 \cos \left (d x +c \right )^{4}-41200 \cos \left (d x +c \right )^{3}-20085 \cos \left (d x +c \right )^{2}+17304 \cos \left (d x +c \right )+10815\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}+\left (4210 \cos \left (d x +c \right )^{6}+9630 \cos \left (d x +c \right )^{5}-26980 \cos \left (d x +c \right )^{4}-8860 \cos \left (d x +c \right )^{3}+29010 \cos \left (d x +c \right )^{2}+3070 \cos \left (d x +c \right )-10080\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}\right )}{7680 d \left (1+\cos \left (d x +c \right )\right )^{2}}\)	\(398\)

Input:

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

Output:

1/7680/d*a*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))^2*((7680*cos(d*x+c)^2+1 
5360*cos(d*x+c)+7680)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(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)+cs 
c(d*x+c)^2-1)^(1/2))+(2640*cos(d*x+c)^2+5280*cos(d*x+c)+2640)*(-2*cos(d*x+ 
c)/(1+cos(d*x+c)))^(1/2)*ln((-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-cot(d*x+c 
)+csc(d*x+c))+2^(1/2)*(9289*cos(d*x+c)^6+30040*cos(d*x+c)^5+6125*cos(d*x+c 
)^4-41200*cos(d*x+c)^3-20085*cos(d*x+c)^2+17304*cos(d*x+c)+10815)*(-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+(4210*cos(d*x+c)^6+9630*cos(d*x+c)^5-26980*cos(d*x+c)^4-8860 
*cos(d*x+c)^3+29010*cos(d*x+c)^2+3070*cos(d*x+c)-10080)*cot(d*x+c)*csc(d*x 
+c)^4)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 688, normalized size of antiderivative = 2.98 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/480*(165*sqrt(1/2)*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c 
) + a)*sqrt(-a)*log(-(4*sqrt(1/2)*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) + 240*(a*cos(d*x 
 + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*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) - 2*(449*a*cos(d*x + c)^4 - 351*a*cos(d*x + c)^3 - 365*a 
*cos(d*x + c)^2 + 315*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + 
c)))/((d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + 
 c)), -1/240*(165*sqrt(1/2)*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d 
*x + c) + a)*sqrt(a)*arctan(2*sqrt(1/2)*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) + 240*(a*cos(d*x + 
 c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*sqrt(a)*arctan(2*sqrt(a)*sq 
rt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d 
*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + (449*a*cos(d*x + c)^4 - 35 
1*a*cos(d*x + c)^3 - 365*a*cos(d*x + c)^2 + 315*a*cos(d*x + c))*sqrt((a*co 
s(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*c 
os(d*x + c) + d)*sin(d*x + c))]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (198) = 396\).

Time = 1.22 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.25 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/960*(165*sqrt(2)*sqrt(-a)*a*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a 
*tan(1/2*d*x + 1/2*c)^2 + a))^2)*sgn(cos(d*x + c)) + 30*sqrt(2)*sqrt(-a*ta 
n(1/2*d*x + 1/2*c)^2 + a)*a*sgn(cos(d*x + c))*tan(1/2*d*x + 1/2*c) + 960*s 
qrt(-a)*a^2*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 
))*sgn(cos(d*x + c))/abs(a) + 32*sqrt(2)*(60*(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)*a^2*sgn(cos(d*x + c)) 
- 195*(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^3*sgn(cos(d*x + c)) + 275*(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^4*sgn(cos(d*x + c)) - 17 
5*(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^5*sgn(cos(d*x + c)) + 47*sqrt(-a)*a^6*sgn(cos(d*x + c)))/((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
 

Mupad [F(-1)]

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

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

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

Output:

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