Integrand size = 21, antiderivative size = 144 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {7}{4 a^2 d \sqrt {a+a \sec (c+d x)}} \]
2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-1/5/d/(a+a*sec(d*x+c)) ^(5/2)-1/2/a/d/(a+a*sec(d*x+c))^(3/2)-1/8*arctanh(1/2*(a+a*sec(d*x+c))^(1/ 2)*2^(1/2)/a^(1/2))/a^(5/2)/d*2^(1/2)-7/4/a^2/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 = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {1}{2} (1+\sec (c+d x))\right )-2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1+\sec (c+d x)\right )}{5 d (a (1+\sec (c+d x)))^{5/2}} \]
(Hypergeometric2F1[-5/2, 1, -3/2, (1 + Sec[c + d*x])/2] - 2*Hypergeometric 2F1[-5/2, 1, -3/2, 1 + Sec[c + d*x]])/(5*d*(a*(1 + Sec[c + d*x]))^(5/2))
Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 25, 4368, 25, 27, 96, 25, 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.
| \(\displaystyle \int \frac {\cot (c+d x)}{(a \sec (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right ) \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4368 |
| \(\displaystyle \frac {a^2 \int -\frac {\cos (c+d x)}{a (1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \int \frac {\cos (c+d x)}{a (1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {\cos (c+d x)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 96 |
| \(\displaystyle -\frac {a \left (\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}-\frac {\int -\frac {a \cos (c+d x) (2-\sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {a \cos (c+d x) (2-\sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a^2}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\cos (c+d x) (2-\sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {3 a \cos (c+d x) (4-3 \sec (c+d x))}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{3 a^2}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\int \frac {\cos (c+d x) (4-3 \sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\int \frac {a \cos (c+d x) (8-7 \sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {7}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\int \frac {\cos (c+d x) (8-7 \sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {7}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+8 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {7}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {2 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {16 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {7}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {16 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {7}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle -\frac {a \left (\frac {\frac {\frac {\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {16 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {7}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {1}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}+\frac {1}{5 a (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
-((a*(1/(5*a*(a + a*Sec[c + d*x])^(5/2)) + (1/(a*(a + a*Sec[c + d*x])^(3/2 )) + (((-16*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sqrt[a] + (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]]))/(2*a))/(2*a)))/d)
3.2.98.3.1 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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S imp[1/((b*e - a*f)*(d*e - c*f)) Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -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]
Leaf count of result is larger than twice the leaf count of optimal. \(377\) vs. \(2(115)=230\).
Time = 1.73 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.62
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \left (-15 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}}+15 \left (1-\cos \left (d x +c \right )\right )^{6} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{6}+21 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {5}{2}}-87 \left (1-\cos \left (d x +c \right )\right )^{4} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{4}-35 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}}+269 \left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{2}+840 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}{2}\right )+105 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right )-932 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right )}{840 d \,a^{3}}\) | \(378\) |
-1/840/d/a^3*(-2*a/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)*((1-cos(d*x+c) )^2*csc(d*x+c)^2-1)^(1/2)*(-15*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(7/2)+15* (1-cos(d*x+c))^6*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)*csc(d*x+c)^6+21*( (1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(5/2)-87*(1-cos(d*x+c))^4*((1-cos(d*x+c)) ^2*csc(d*x+c)^2-1)^(1/2)*csc(d*x+c)^4-35*((1-cos(d*x+c))^2*csc(d*x+c)^2-1) ^(3/2)+269*(1-cos(d*x+c))^2*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2)*csc(d* x+c)^2+840*2^(1/2)*arctan(1/2*2^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1 /2))+105*arctan(1/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)^(1/2))-932*((1-cos(d*x +c))^2*csc(d*x+c)^2-1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (115) = 230\).
Time = 0.36 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.98 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [\frac {5 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\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 ) + 40 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 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 ) - 4 \, {\left (49 \, \cos \left (d x + c\right )^{3} + 80 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{80 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, \frac {5 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\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 ) - 40 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 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 (49 \, \cos \left (d x + c\right )^{3} + 80 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{40 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\right ] \]
[1/80*(5*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)* sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*co s(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 40*(cos(d*x + c)^ 3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 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)/c os(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(49*cos(d*x + c)^3 + 80*cos(d*x + c)^2 + 35*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*c os(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d), 1/ 40*(5*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqr t(-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)) - 40*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3 *cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/co s(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(49*cos(d*x + c)^3 + 80*cos(d*x + c)^2 + 35*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c )))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)]
\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.93 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.43 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {5 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {80 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {\sqrt {2} {\left ({\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{20} + 5 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{21} + 35 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{22}\right )}}{a^{25} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{40 \, d} \]
1/40*(5*sqrt(2)*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt (-a)*a^2*sgn(cos(d*x + c))) - 80*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a^2*sgn(cos(d*x + c))) - sqrt(2)*((a*tan (1/2*d*x + 1/2*c)^2 - a)^2*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^20 + 5*(- a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2)*a^21 + 35*sqrt(-a*tan(1/2*d*x + 1/2*c) ^2 + a)*a^22)/(a^25*sgn(cos(d*x + c))))/d
Timed out. \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\mathrm {cot}\left (c+d\,x\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]