Integrand size = 23, antiderivative size = 193 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {107 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 d \sqrt {a+a \sec (c+d x)}} \]
43/96*a^2/d/(a+a*sec(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)+2*arctanh((a+ a*sec(d*x+c))^(1/2)/a^(1/2))*a^(1/2)/d-107/128*arctanh(1/2*(a+a*sec(d*x+c) )^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*a^(1/2)/d-21/64*a/d/(a+a*sec(d*x+c))^(1/2 )
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.53 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\cot ^4(c+d x) \left (-2 \left (57+32 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2-45 \sec (c+d x)\right )+107 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2\right ) \sqrt {a (1+\sec (c+d x))}}{96 d} \]
(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)
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 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {107 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {256 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {21}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {43}{3 a (a \sec (c+d x)+a)^{3/2}}\right )+\frac {15}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}\right )}{d}\) |
-((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)
3.2.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]
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]
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])
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])
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]
Time = 3.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (321 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+768 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+410 \cot \left (d x +c \right )^{4}-142 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )-298 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}+126 \csc \left (d x +c \right )^{3} \cot \left (d x +c \right )\right )}{384 d}\) | \(173\) |
-1/384/d*(a*(1+sec(d*x+c)))^(1/2)*(321*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1) )^(1/2)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+768*arctan( (-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+410 *cot(d*x+c)^4-142*cot(d*x+c)^3*csc(d*x+c)-298*cot(d*x+c)^2*csc(d*x+c)^2+12 6*csc(d*x+c)^3*cot(d*x+c))
Time = 0.37 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.74 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\left [\frac {384 \, {\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 ) + 321 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {2}\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 ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) - 4 \, {\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 )}}}{768 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}}, \frac {321 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {2}\right )} \sqrt {-a} \arctan \left (\frac {\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 )}{a \cos \left (d x + c\right ) + a}\right ) - 384 \, {\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 ) - 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 )}}\right ] \]
[1/768*(384*(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 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 321*(sqrt(2)*cos(d*x + c)^ 4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sq rt((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)) - 4*(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), 1/384*(321*(sqrt(2)*cos(d*x + c )^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(-a)*arctan(sqrt(2)*sqrt(-a) *sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a) ) - 384*(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)) - 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)]
\[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \cot ^{5}{\left (c + d x \right )}\, dx \]
Timed out. \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{5} \,d x } \]
Timed out. \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^5\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]