\(\int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx\) [152]

Optimal result

Integrand size = 23, antiderivative size = 171 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {71 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d}+\frac {7 a^2}{32 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}} \] Output:

2*a^(3/2)*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/d-71/64*2^(1/2)*a^(3/2)* 
arctanh(1/2*(a+a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d+7/32*a^2/d/(a+a*sec( 
d*x+c))^(1/2)-1/4*a^2/d/(1-sec(d*x+c))^2/(a+a*sec(d*x+c))^(1/2)-13/16*a^2/ 
d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.61 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {a^2 \left (-34+71 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+26 \sec (c+d x)\right )}{32 d (-1+\sec (c+d x))^2 \sqrt {a (1+\sec (c+d x))}} \] Input:

Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2),x]
 

Output:

(a^2*(-34 + 71*Hypergeometric2F1[-1/2, 1, 1/2, (1 + Sec[c + d*x])/2]*(-1 + 
 Sec[c + d*x])^2 - 64*Hypergeometric2F1[-1/2, 1, 1/2, 1 + Sec[c + d*x]]*(- 
1 + Sec[c + d*x])^2 + 26*Sec[c + d*x]))/(32*d*(-1 + Sec[c + d*x])^2*Sqrt[a 
*(1 + Sec[c + d*x])])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 25, 4368, 25, 27, 114, 27, 168, 27, 169, 27, 174, 73, 219, 220}

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.

\(\displaystyle \int \cot ^5(c+d x) (a \sec (c+d x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\cot \left (c+d x+\frac {\pi }{2}\right )^5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^{3/2}}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5}dx\)

\(\Big \downarrow \) 4368

\(\displaystyle \frac {a^6 \int -\frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^6 \int \frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \int \frac {\cos (c+d x)}{(1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 114

\(\displaystyle -\frac {a^3 \left (\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}-\frac {\int -\frac {a \cos (c+d x) (5 \sec (c+d x)+8)}{2 (1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{4 a}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \int \frac {\cos (c+d x) (5 \sec (c+d x)+8)}{(1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}-\frac {\int -\frac {a \cos (c+d x) (39 \sec (c+d x)+32)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {\cos (c+d x) (39 \sec (c+d x)+32)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {a \cos (c+d x) (7 \sec (c+d x)+64)}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}-\frac {7}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {\cos (c+d x) (7 \sec (c+d x)+64)}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}-\frac {7}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {71 \int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+64 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}-\frac {7}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {142 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {128 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}-\frac {7}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {128 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {71 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}-\frac {7}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 220

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {71 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {128 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}-\frac {7}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {13}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

Input:

Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2),x]
 

Output:

-((a^3*(1/(4*a*(1 - Sec[c + d*x])^2*Sqrt[a + a*Sec[c + d*x]]) + (13/(2*a*( 
1 - Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]) + (((-128*ArcTanh[Sqrt[a + a*S 
ec[c + d*x]]/Sqrt[a]])/Sqrt[a] + (71*Sqrt[2]*ArcTanh[Sqrt[a + a*Sec[c + d* 
x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[a])/(2*a) - 7/(a*Sqrt[a + a*Sec[c + d*x]]))/4 
)/8))/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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 219

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

rule 220

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4368

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n 
_), x_Symbol] :> Simp[-(d*b^(m - 1))^(-1)   Subst[Int[(-a + b*x)^((m - 1)/2 
)*((a + b*x)^((m - 1)/2 + n)/x), x], x, Csc[c + d*x]], x] /; FreeQ[{a, b, c 
, d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[n]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(142)=284\).

Time = 1.11 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.99

method	result	size
default	\(-\frac {a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (974 \cos \left (d x +c \right )^{5}-10270 \cos \left (d x +c \right )^{4}-18628 \cos \left (d x +c \right )^{3}-1420 \cos \left (d x +c \right )^{2}+10934 \cos \left (d x +c \right )+4970\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}+\left (6720 \cos \left (d x +c \right )^{2}+13440 \cos \left (d x +c \right )+6720\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (7455 \cos \left (d x +c \right )^{2}+14910 \cos \left (d x +c \right )+7455\right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (7618 \cos \left (d x +c \right )^{5}-7998 \cos \left (d x +c \right )^{4}-6788 \cos \left (d x +c \right )^{3}+5628 \cos \left (d x +c \right )^{2}+3010 \cos \left (d x +c \right )-1470\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}\right )}{6720 d \left (1+\cos \left (d x +c \right )\right )^{2}}\)	\(340\)

Input:

int(cot(d*x+c)^5*(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6720/d*a*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))^2*((974*cos(d*x+c)^5-1 
0270*cos(d*x+c)^4-18628*cos(d*x+c)^3-1420*cos(d*x+c)^2+10934*cos(d*x+c)+49 
70)*2^(1/2)*(-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)^3+(6720*cos(d*x+c)^2+13440*cos(d*x+c)+6720 
)*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*(-2*cos 
(d*x+c)/(1+cos(d*x+c)))^(1/2)+(7455*cos(d*x+c)^2+14910*cos(d*x+c)+7455)*ar 
ctan(1/2*2^(1/2)/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*(-2*cos(d*x+c)/(1+cos 
(d*x+c)))^(1/2)+(7618*cos(d*x+c)^5-7998*cos(d*x+c)^4-6788*cos(d*x+c)^3+562 
8*cos(d*x+c)^2+3010*cos(d*x+c)-1470)*cot(d*x+c)*csc(d*x+c)^3)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.33 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

[1/64*(71*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) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 32*(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)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + 
c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 2*(27*a*cos(d*x + c)^3 - 1 
2*a*cos(d*x + c)^2 - 7*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), 1/32*(71 
*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)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*c 
os(d*x + c)/(a*cos(d*x + c) + a)) - 32*(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)/(2*a*cos(d*x + c) + a)) - (27*a*cos(d*x + c) 
^3 - 12*a*cos(d*x + c)^2 - 7*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)]
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**5*(a+a*sec(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.23 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\frac {\frac {71 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt {-a}} - \frac {128 \, a^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt {-a}} - 8 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - \frac {17 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 15 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \] Input:

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

1/64*(71*sqrt(2)*a^2*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))* 
sgn(cos(d*x + c))/sqrt(-a) - 128*a^2*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d* 
x + 1/2*c)^2 + a)/sqrt(-a))*sgn(cos(d*x + c))/sqrt(-a) - 8*sqrt(2)*sqrt(-a 
*tan(1/2*d*x + 1/2*c)^2 + a)*a*sgn(cos(d*x + c)) - (17*sqrt(2)*(-a*tan(1/2 
*d*x + 1/2*c)^2 + a)^(3/2)*a^2*sgn(cos(d*x + c)) - 15*sqrt(2)*sqrt(-a*tan( 
1/2*d*x + 1/2*c)^2 + a)*a^3*sgn(cos(d*x + c)))/(a^2*tan(1/2*d*x + 1/2*c)^4 
))/d
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \] Input:

int(cot(c + d*x)^5*(a + a/cos(c + d*x))^(3/2),x)
 

Output:

int(cot(c + d*x)^5*(a + a/cos(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{5} \sec \left (d x +c \right )d x +\int \sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{5}d x \right ) \] Input:

int(cot(d*x+c)^5*(a+a*sec(d*x+c))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(sec(c + d*x) + 1)*cot(c + d*x)**5*sec(c + d*x),x) + in 
t(sqrt(sec(c + d*x) + 1)*cot(c + d*x)**5,x))