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

Optimal result

Integrand size = 23, antiderivative size = 226 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {11 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} d}-\frac {21 a \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{16 d}+\frac {5 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{24 d}+\frac {3 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{20 a d}-\frac {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d} \]

[Out]

-2*a^(3/2)*arctan(a^(1/2)*tan(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-1/4*cos(d*x+c)*cot(d*x+c)^5*sec(1/2*d*x+1/2*c)^2*(a+a*sec(d*x+c))^(5/
2)/a/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))/d*2^(1/2)-21/16*a*cot(d*x+c
)*(a+a*sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3972, 483, 597, 536, 209} \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {11 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} d}+\frac {3 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{20 a d}+\frac {5 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}-\frac {21 a \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{16 d}-\frac {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{4 a d} \]

[In]

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

[Out]

(-2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (11*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d
*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*d) - (21*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(16*d)
+ (5*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(24*d) + (3*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(20*a*
d) - (Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(5/2))/(4*a*d)

Rule 209

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

Rule 483

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

Rule 536

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

Rule 597

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

Rule 3972

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-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] && IntegerQ[n - 1/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a d} \\ & = -\frac {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d}-\frac {\text {Subst}\left (\int \frac {-3 a-7 a^2 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2 a^2 d} \\ & = \frac {3 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{20 a d}-\frac {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d}+\frac {\text {Subst}\left (\int \frac {25 a^2-15 a^3 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{20 a^2 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 {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d}-\frac {\text {Subst}\left (\int \frac {315 a^3+75 a^4 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{120 a^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 {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d}+\frac {\text {Subst}\left (\int \frac {795 a^4+315 a^5 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{240 a^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 {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d}-\frac {\left (11 a^2\right ) \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{16 d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\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 {\cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{4 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.46 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {(a (1+\sec (c+d x)))^{3/2} \left (165 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}+2 \sqrt {2} \left (-240 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}+\frac {\sqrt {\frac {1}{1+\cos (c+d x)}} \csc ^5(c+d x) (281-279 \sec (c+d x)+\cos (2 (c+d x)) (-449+351 \sec (c+d x)))}{\sec ^{\frac {3}{2}}(c+d x)}\right )\right )}{960 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

((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)]*S
qrt[1 + Sec[c + d*x]] + 2*Sqrt[2]*(-240*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))

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {a \left (-165 \sin \left (d x +c \right )^{5} \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+960 \sin \left (d x +c \right )^{5} \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+898 \cos \left (d x +c \right )^{5}+196 \cos \left (d x +c \right )^{4}-1432 \cos \left (d x +c \right )^{3}-100 \cos \left (d x +c \right )^{2}+630 \cos \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \csc \left (d x +c \right )^{5}}{480 d}\)	\(227\)

[In]

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

[Out]

-1/480/d*a*(-165*sin(d*x+c)^5*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c
)^2-1)^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+960*sin(d*x+c)^5*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+
c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+898*cos(d*x+c)^5+196*cos(d*x+c)^4-1432*cos(d*x+c)
^3-100*cos(d*x+c)^2+630*cos(d*x+c))*(a*(1+sec(d*x+c)))^(1/2)*csc(d*x+c)^5

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 708, normalized size of antiderivative = 3.13 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\left [\frac {165 \, {\left (\sqrt {2} a \cos \left (d x + c\right )^{3} - \sqrt {2} a \cos \left (d x + c\right )^{2} - \sqrt {2} a \cos \left (d x + c\right ) + \sqrt {2} a\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 480 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, {\left (449 \, a \cos \left (d x + c\right )^{4} - 351 \, a \cos \left (d x + c\right )^{3} - 365 \, a \cos \left (d x + c\right )^{2} + 315 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{960 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )}, -\frac {480 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 165 \, {\left (\sqrt {2} a \cos \left (d x + c\right )^{3} - \sqrt {2} a \cos \left (d x + c\right )^{2} - \sqrt {2} a \cos \left (d x + c\right ) + \sqrt {2} a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 2 \, {\left (449 \, a \cos \left (d x + c\right )^{4} - 351 \, a \cos \left (d x + c\right )^{3} - 365 \, a \cos \left (d x + c\right )^{2} + 315 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{480 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )}\right ] \]

[In]

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

[Out]

[1/960*(165*(sqrt(2)*a*cos(d*x + c)^3 - sqrt(2)*a*cos(d*x + c)^2 - sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sqrt(-a
)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c
)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 480*(a*cos(d*x + c)^3 - a*co
s(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))*sqr
t(-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*(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*co
s(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)), -1/4
80*(480*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*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*cos(d*x + c) - a))*sin(d*x + c) + 165*
(sqrt(2)*a*cos(d*x + c)^3 - sqrt(2)*a*cos(d*x + c)^2 - sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*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) + 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))]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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