Integrand size = 25, antiderivative size = 268 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{5/2} f}-\frac {\left (8 a^2+36 a b+63 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} f}+\frac {b \left (12 a^2+39 a b-8 b^2\right )}{24 a (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {(4 a+11 b) \cot ^2(e+f x)}{8 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+15 a^2 b-32 a b^2-8 b^3\right )}{8 a^2 (a+b)^4 f \sqrt {a+b \sec ^2(e+f x)}} \]
arctanh((a+b*sec(f*x+e)^2)^(1/2)/a^(1/2))/a^(5/2)/f-1/8*(8*a^2+36*a*b+63*b ^2)*arctanh((a+b*sec(f*x+e)^2)^(1/2)/(a+b)^(1/2))/(a+b)^(9/2)/f+1/24*b*(12 *a^2+39*a*b-8*b^2)/a/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(3/2)+1/8*(4*a+11*b)*cot (f*x+e)^2/(a+b)^2/f/(a+b*sec(f*x+e)^2)^(3/2)-1/4*cot(f*x+e)^4/(a+b)/f/(a+b *sec(f*x+e)^2)^(3/2)+1/8*b*(4*a^3+15*a^2*b-32*a*b^2-8*b^3)/a^2/(a+b)^4/f/( a+b*sec(f*x+e)^2)^(1/2)
\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Time = 0.52 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4627, 25, 354, 114, 27, 168, 27, 169, 27, 169, 27, 174, 73, 221}
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.
| \(\displaystyle \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^5 \left (a+b \sec (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4627 |
| \(\displaystyle \frac {\int -\frac {\cos (e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\cos (e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {\int \frac {\cos (e+f x)}{\left (1-\sec ^2(e+f x)\right )^3 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\cos (e+f x) \left (7 b \sec ^2(e+f x)+4 (a+b)\right )}{2 \left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{2 (a+b)}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\cos (e+f x) \left (7 b \sec ^2(e+f x)+4 (a+b)\right )}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {\frac {\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\cos (e+f x) \left (8 (a+b)^2+5 b (4 a+11 b) \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{a+b}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cos (e+f x) \left (8 (a+b)^2+5 b (4 a+11 b) \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {\frac {2 \int \frac {3 \cos (e+f x) \left (8 (a+b)^3+b \left (12 a^2+39 b a-8 b^2\right ) \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec ^2(e+f x)}{3 a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cos (e+f x) \left (8 (a+b)^3+b \left (12 a^2+39 b a-8 b^2\right ) \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {2 \int \frac {\cos (e+f x) \left (8 (a+b)^4+b \left (4 a^3+15 b a^2-32 b^2 a-8 b^3\right ) \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (4 a^3+15 a^2 b-32 a b^2-8 b^3\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {\int \frac {\cos (e+f x) \left (8 (a+b)^4+b \left (4 a^3+15 b a^2-32 b^2 a-8 b^3\right ) \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (4 a^3+15 a^2 b-32 a b^2-8 b^3\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {a^2 \left (8 a^2+36 a b+63 b^2\right ) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)+8 (a+b)^4 \int \frac {\cos (e+f x)}{\sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (4 a^3+15 a^2 b-32 a b^2-8 b^3\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {\frac {2 a^2 \left (8 a^2+36 a b+63 b^2\right ) \int \frac {1}{\frac {a+b}{b}-\frac {\sec ^4(e+f x)}{b}}d\sqrt {b \sec ^2(e+f x)+a}}{b}+\frac {16 (a+b)^4 \int \frac {1}{\frac {\sec ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \sec ^2(e+f x)+a}}{b}}{a (a+b)}-\frac {2 b \left (4 a^3+15 a^2 b-32 a b^2-8 b^3\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {\frac {2 a^2 \left (8 a^2+36 a b+63 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-\frac {16 (a+b)^4 \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a+b)}-\frac {2 b \left (4 a^3+15 a^2 b-32 a b^2-8 b^3\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}-\frac {2 b \left (12 a^2+39 a b-8 b^2\right )}{3 a (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {4 a+11 b}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{4 (a+b)}+\frac {1}{2 (a+b) \left (1-\sec ^2(e+f x)\right )^2 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\) |
-1/2*(1/(2*(a + b)*(1 - Sec[e + f*x]^2)^2*(a + b*Sec[e + f*x]^2)^(3/2)) + ((4*a + 11*b)/((a + b)*(1 - Sec[e + f*x]^2)*(a + b*Sec[e + f*x]^2)^(3/2)) + ((-2*b*(12*a^2 + 39*a*b - 8*b^2))/(3*a*(a + b)*(a + b*Sec[e + f*x]^2)^(3 /2)) + (((-16*(a + b)^4*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/Sqrt[ a] + (2*a^2*(8*a^2 + 36*a*b + 63*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/S qrt[a + b]])/Sqrt[a + b])/(a*(a + b)) - (2*b*(4*a^3 + 15*a^2*b - 32*a*b^2 - 8*b^3))/(a*(a + b)*Sqrt[a + b*Sec[e + f*x]^2]))/(a*(a + b)))/(2*(a + b)) )/(4*(a + b)))/f
3.5.33.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Si mp[1/f Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x), x] , x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[( m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || Integers Q[2*n, p])
Leaf count of result is larger than twice the leaf count of optimal. \(77708\) vs. \(2(238)=476\).
Time = 10.43 (sec) , antiderivative size = 77709, normalized size of antiderivative = 289.96
Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (238) = 476\).
Time = 18.70 (sec) , antiderivative size = 4751, normalized size of antiderivative = 17.73 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/96*(12*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5) *cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos( f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^ 5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2 *b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b ^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b )/cos(f*x + e)^2)) + 3*((8*a^7 + 36*a^6*b + 63*a^5*b^2)*cos(f*x + e)^8 + 8 *a^5*b^2 + 36*a^4*b^3 + 63*a^3*b^4 - 2*(8*a^7 + 28*a^6*b + 27*a^5*b^2 - 63 *a^4*b^3)*cos(f*x + e)^6 + (8*a^7 + 4*a^6*b - 73*a^5*b^2 - 216*a^4*b^3 + 6 3*a^3*b^4)*cos(f*x + e)^4 + 2*(8*a^6*b + 28*a^5*b^2 + 27*a^4*b^3 - 63*a^3* b^4)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e) ^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2 ))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((18*a^7 + 69*a^6*b + 51*a ^5*b^2 + 104*a^4*b^3 + 136*a^3*b^4 + 32*a^2*b^5)*cos(f*x + e)^8 - (12*a^7 + 21*a^6*b - 93*a^5*b^2 + 106*a^4*b^3 + 176*a^3*b^4 - 56*a^2*b^5 - 24*a*b^ 6)*cos(f*x + e)^6 - (24*a^6*b + 96*a^5*b^2 - 83*a^4*b^3 + 5*a^3*b^4 + 2...
\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \]