\(\int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\) [140]

Optimal result

Integrand size = 23, antiderivative size = 193 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {107 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 d \sqrt {a+a \sec (c+d x)}} \] Output:

2*a^(1/2)*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/d-107/128*2^(1/2)*a^(1/2 
)*arctanh(1/2*(a+a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d+43/96*a^2/d/(a+a*s 
ec(d*x+c))^(3/2)-1/4*a^2/d/(1-sec(d*x+c))^2/(a+a*sec(d*x+c))^(3/2)-15/16*a 
^2/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(3/2)-21/64*a/d/(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 = 102, normalized size of antiderivative = 0.53 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\cot ^4(c+d x) \left (-2 \left (57+32 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2-45 \sec (c+d x)\right )+107 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2\right ) \sqrt {a (1+\sec (c+d x))}}{96 d} \] Input:

Integrate[Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]],x]
 

Output:

(Cot[c + d*x]^4*(-2*(57 + 32*Hypergeometric2F1[-3/2, 1, -1/2, 1 + Sec[c + 
d*x]]*(-1 + Sec[c + d*x])^2 - 45*Sec[c + d*x]) + 107*Hypergeometric2F1[-3/ 
2, 1, -1/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2)*Sqrt[a*(1 + Sec[c 
 + d*x])])/(96*d)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 25, 4368, 25, 27, 114, 27, 168, 27, 169, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4368

\(\displaystyle \frac {a^6 \int -\frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^6 \int \frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 114

\(\displaystyle -\frac {a^3 \left (\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}-\frac {\int -\frac {a \cos (c+d x) (7 \sec (c+d x)+8)}{2 (1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{4 a}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \int \frac {\cos (c+d x) (7 \sec (c+d x)+8)}{(1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac {\int -\frac {a \cos (c+d x) (75 \sec (c+d x)+32)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {\cos (c+d x) (75 \sec (c+d x)+32)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {3 a \cos (c+d x) (43 \sec (c+d x)+64)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{3 a^2}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {\cos (c+d x) (43 \sec (c+d x)+64)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {a \cos (c+d x) (128-21 \sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {\cos (c+d x) (128-21 \sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {107 \int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+128 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {214 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {256 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {256 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {107 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

\(\Big \downarrow \) 220

\(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {107 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {256 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\)

Input:

Int[Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]],x]
 

Output:

-((a^3*(1/(4*a*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(3/2)) + (15/(2*a 
*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2)) + (-43/(3*a*(a + a*Sec[c + 
 d*x])^(3/2)) + (((-256*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sqrt[a] 
 + (107*Sqrt[2]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[ 
a])/(2*a) + 21/(a*Sqrt[a + a*Sec[c + d*x]]))/(2*a))/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. \(398\) vs. \(2(160)=320\).

Time = 0.83 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.07

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (16834 \cos \left (d x +c \right )^{6}-28336 \cos \left (d x +c \right )^{5}-122638 \cos \left (d x +c \right )^{4}-83888 \cos \left (d x +c \right )^{3}+43014 \cos \left (d x +c \right )^{2}+71904 \cos \left (d x +c \right )+22470\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}+\left (40320 \cos \left (d x +c \right )^{3}+120960 \cos \left (d x +c \right )^{2}+120960 \cos \left (d x +c \right )+40320\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right )+\left (33705 \cos \left (d x +c \right )^{3}+101115 \cos \left (d x +c \right )^{2}+101115 \cos \left (d x +c \right )+33705\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\left (76718 \cos \left (d x +c \right )^{6}+23900 \cos \left (d x +c \right )^{5}-101806 \cos \left (d x +c \right )^{4}-95160 \cos \left (d x +c \right )^{3}+29778 \cos \left (d x +c \right )^{2}+53340 \cos \left (d x +c \right )+13230\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}\right )}{40320 d \left (1+\cos \left (d x +c \right )\right ) \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right )}\)	\(399\)

Input:

int(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/40320/d*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(cos(d*x+c)^2+2*cos(d*x 
+c)+1)*((16834*cos(d*x+c)^6-28336*cos(d*x+c)^5-122638*cos(d*x+c)^4-83888*c 
os(d*x+c)^3+43014*cos(d*x+c)^2+71904*cos(d*x+c)+22470)*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)*cs 
c(d*x+c)^3+(40320*cos(d*x+c)^3+120960*cos(d*x+c)^2+120960*cos(d*x+c)+40320 
)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos( 
d*x+c)/(1+cos(d*x+c)))^(1/2))+(33705*cos(d*x+c)^3+101115*cos(d*x+c)^2+1011 
15*cos(d*x+c)+33705)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*2^(1/ 
2)/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+(76718*cos(d*x+c)^6+23900*cos(d*x+c 
)^5-101806*cos(d*x+c)^4-95160*cos(d*x+c)^3+29778*cos(d*x+c)^2+53340*cos(d* 
x+c)+13230)*cot(d*x+c)*csc(d*x+c)^3)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.68 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\left [\frac {321 \, \sqrt {\frac {1}{2}} {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt {a} \log \left (-\frac {4 \, \sqrt {\frac {1}{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 ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) + 192 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 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 )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 2 \, {\left (205 \, \cos \left (d x + c\right )^{4} - 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{384 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}}, \frac {321 \, \sqrt {\frac {1}{2}} {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {\frac {1}{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 )}{a \cos \left (d x + c\right ) + a}\right ) - 192 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\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 )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - {\left (205 \, \cos \left (d x + c\right )^{4} - 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{192 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}}\right ] \] Input:

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

[1/384*(321*sqrt(1/2)*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*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)) + 192*(cos(d*x + c)^4 - 2*cos 
(d*x + c)^2 + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + c 
os(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x 
 + c) - a) - 2*(205*cos(d*x + c)^4 - 71*cos(d*x + c)^3 - 149*cos(d*x + c)^ 
2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + 
 c)^4 - 2*d*cos(d*x + c)^2 + d), 1/192*(321*sqrt(1/2)*(cos(d*x + c)^4 - 2* 
cos(d*x + c)^2 + 1)*sqrt(-a)*arctan(2*sqrt(1/2)*sqrt(-a)*sqrt((a*cos(d*x + 
 c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 192*(cos(d*x + 
 c)^4 - 2*cos(d*x + c)^2 + 1)*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)) - (205*cos(d*x 
 + c)^4 - 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt(( 
a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 
+ d)]
 

Sympy [F]

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

integrate(cot(d*x+c)**5*(a+a*sec(d*x+c))**(1/2),x)
 

Output:

Integral(sqrt(a*(sec(c + d*x) + 1))*cot(c + d*x)**5, x)
 

Maxima [F(-1)]

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

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=-\frac {\sqrt {2} {\left (\frac {384 \, \sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {321 \, a \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {8 \, {\left ({\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{2} + 15 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{3}\right )}}{a^{3}} + \frac {3 \, {\left (21 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a - 19 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{2}\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{384 \, d} \] Input:

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

-1/384*sqrt(2)*(384*sqrt(2)*a*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2 
*c)^2 + a)/sqrt(-a))/sqrt(-a) - 321*a*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^ 
2 + a)/sqrt(-a))/sqrt(-a) + 8*((-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2)*a^2 + 
 15*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^3)/a^3 + 3*(21*(-a*tan(1/2*d*x + 
 1/2*c)^2 + a)^(3/2)*a - 19*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^2)/(a^2* 
tan(1/2*d*x + 1/2*c)^4))*sgn(cos(d*x + c))/d
 

Mupad [F(-1)]

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

int(cot(c + d*x)^5*(a + a/cos(c + d*x))^(1/2),x)
 

Output:

int(cot(c + d*x)^5*(a + a/cos(c + d*x))^(1/2), x)
 

Reduce [F]

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

int(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x)
 

Output:

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