\(\int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx\) [170]

Optimal result

Integrand size = 23, antiderivative size = 176 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d} \] Output:

-2*a^(5/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+1/8*a^(5/2) 
*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-7 
/4*a^2*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d+1/2*a*cot(d*x+c)^3*(a+a*sec(d*x 
+c))^(3/2)/d-1/5*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/d
 

Mathematica [A] (warning: unable to verify)

Time = 5.64 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.15 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {\sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (\frac {1}{40} \cos ^3(c+d x) (160 \cos (c+d x)-7 (17+7 \cos (2 (c+d x)))) \csc ^5\left (\frac {1}{2} (c+d x)\right )+\frac {\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )-8 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {1+\sec (c+d x)}}{\sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {5}{2}}(c+d x)}\right )}{32 d} \] Input:

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

Output:

(Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*((Cos[c + d*x]^3*(160*Cos 
[c + d*x] - 7*(17 + 7*Cos[2*(c + d*x)]))*Csc[(c + d*x)/2]^5)/40 + ((ArcSin 
[Tan[(c + d*x)/2]] - 8*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/Sqrt[Cos[c + d*x]/( 
1 + Cos[c + d*x])]])*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2 
]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2))))/ 
(32*d)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4375, 382, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4375

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

\(\Big \downarrow \) 382

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{6} \int \frac {3 a \cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+7\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \int \frac {\cot ^2(c+d x) (\sec (c+d x) a+a) \left (\frac {3 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+7\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )}{d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {2 \left (\frac {1}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {7}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} \int \frac {a \left (\frac {7 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+15\right )}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {1}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {7}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \int \frac {\frac {7 a \tan ^2(c+d x)}{\sec (c+d x) a+a}+15}{\left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1\right ) \left (\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2\right )}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )\right )}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {2 \left (\frac {1}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {7}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (8 \int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )-\int \frac {1}{\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+2}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )\right )\right )\right )\right )}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{10} \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}-\frac {1}{2} a \left (\frac {1}{2} \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}-\frac {1}{2} a \left (\frac {7}{2} \cot (c+d x) \sqrt {a \sec (c+d x)+a}-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {8 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{\sqrt {a}}\right )\right )\right )\right )}{d}\)

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))/10 - (a*((Cot[c + d*x]^3* 
(a + a*Sec[c + d*x])^(3/2))/2 - (a*(-1/2*(a*((-8*ArcTan[(Sqrt[a]*Tan[c + d 
*x])/Sqrt[a + a*Sec[c + d*x]]])/Sqrt[a] + ArcTan[(Sqrt[a]*Tan[c + d*x])/(S 
qrt[2]*Sqrt[a + a*Sec[c + d*x]])]/(Sqrt[2]*Sqrt[a]))) + (7*Cot[c + d*x]*Sq 
rt[a + a*Sec[c + d*x]])/2))/2))/2))/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 382

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ 
(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b* 
x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m 
+ 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ 
b*c - a*d, 0] && LtQ[m, -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. \(357\) vs. \(2(147)=294\).

Time = 0.88 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.03

\[\frac {a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\sqrt {2}\, \left (883 \cos \left (d x +c \right )^{5}+1557 \cos \left (d x +c \right )^{4}-982 \cos \left (d x +c \right )^{3}-1978 \cos \left (d x +c \right )^{2}+483 \cos \left (d x +c \right )+805\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}+\left (590 \cos \left (d x +c \right )^{5}-260 \cos \left (d x +c \right )^{4}-1920 \cos \left (d x +c \right )^{3}+1420 \cos \left (d x +c \right )^{2}+1010 \cos \left (d x +c \right )-840\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}-480 \sqrt {2}\, \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+60 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )}{480 d \left (1+\cos \left (d x +c \right )\right )}\]

Input:

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

Output:

1/480/d*a^2*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))*(2^(1/2)*(883*cos(d*x+ 
c)^5+1557*cos(d*x+c)^4-982*cos(d*x+c)^3-1978*cos(d*x+c)^2+483*cos(d*x+c)+8 
05)*(-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+(590*cos(d*x+c)^5-260*cos(d*x+c)^4-1920*cos(d*x+ 
c)^3+1420*cos(d*x+c)^2+1010*cos(d*x+c)-840)*cot(d*x+c)*csc(d*x+c)^4-480*2^ 
(1/2)*(1+cos(d*x+c))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(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)-co 
t(d*x+c)))+60*(1+cos(d*x+c))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln((-2*c 
os(d*x+c)/(1+cos(d*x+c)))^(1/2)-cot(d*x+c)+csc(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 636, normalized size of antiderivative = 3.61 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/40*(5*sqrt(1/2)*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*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) + 20*(a^2*cos(d*x + c)^2 - 2*a 
^2*cos(d*x + c) + a^2)*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*(49* 
a^2*cos(d*x + c)^3 - 80*a^2*cos(d*x + c)^2 + 35*a^2*cos(d*x + c))*sqrt((a* 
cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d 
)*sin(d*x + c)), -1/20*(5*sqrt(1/2)*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + 
c) + a^2)*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) + 20*(a^2*cos(d*x + c 
)^2 - 2*a^2*cos(d*x + c) + a^2)*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*cos(d*x + c)^2 + a*c 
os(d*x + c) - a))*sin(d*x + c) + (49*a^2*cos(d*x + c)^3 - 80*a^2*cos(d*x + 
 c)^2 + 35*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d* 
cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (147) = 294\).

Time = 2.14 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.73 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/80*(5*sqrt(2)*sqrt(-a)*a^2*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)) + 80*sqrt(-a)*a^3*log(ab 
s(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) + 4*sqrt(2)*(55*(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^3*sgn(cos(d*x + c)) - 170*(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^4*sg 
n(cos(d*x + c)) + 240*(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^5*sgn(cos(d*x + c)) - 150*(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^6*sgn(co 
s(d*x + c)) + 41*sqrt(-a)*a^7*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. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \] 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]

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

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

Output:

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