Integrand size = 25, antiderivative size = 166 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\left (8 a^2+4 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f}+\frac {(4 a+3 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 a^2 f}-\frac {\cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 a f} \]
-1/8*(8*a^2+4*a*b+3*b^2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(5/2) /f+arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/f/(a-b)^(1/2)+1/8*(4*a+3* b)*cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2)/a^2/f-1/4*cot(f*x+e)^4*(a+b*tan(f *x+e)^2)^(1/2)/a/f
Time = 2.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\left (-8 a^3+4 a^2 b+a b^2+3 b^3\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 a^2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+(-a+b) \cot ^2(e+f x) \left (-4 a-3 b+2 a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}\right )}{8 a^{5/2} (a-b) f} \]
((-8*a^3 + 4*a^2*b + a*b^2 + 3*b^3)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqr t[a]] + Sqrt[a]*(8*a^2*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt [a - b]] + (-a + b)*Cot[e + f*x]^2*(-4*a - 3*b + 2*a*Cot[e + f*x]^2)*Sqrt[ a + b*Tan[e + f*x]^2]))/(8*a^(5/2)*(a - b)*f)
Time = 0.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4153, 354, 114, 27, 168, 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.
| \(\displaystyle \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^5 \sqrt {a+b \tan (e+f x)^2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^5(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {-\frac {\int \frac {\cot ^2(e+f x) \left (3 b \tan ^2(e+f x)+4 a+3 b\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\cot ^2(e+f x) \left (3 b \tan ^2(e+f x)+4 a+3 b\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\cot (e+f x) \left (8 a^2+4 b a+3 b^2+b (4 a+3 b) \tan ^2(e+f x)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a}-\frac {(4 a+3 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\cot (e+f x) \left (8 a^2+4 b a+3 b^2+b (4 a+3 b) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}-\frac {(4 a+3 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {-\frac {-\frac {\left (8 a^2+4 a b+3 b^2\right ) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-8 a^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{2 a}-\frac {(4 a+3 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2 \left (8 a^2+4 a b+3 b^2\right ) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {16 a^2 \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{2 a}-\frac {(4 a+3 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {16 a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 \left (8 a^2+4 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}-\frac {(4 a+3 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\) |
(-1/2*(Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/a - (-1/2*((-2*(8*a^2 + 4*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (16* a^2*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/a - ((4* a + 3*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/(4*a))/(2*f)
3.4.25.3.1 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[(((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[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(1893\) vs. \(2(144)=288\).
Time = 0.97 (sec) , antiderivative size = 1894, normalized size of antiderivative = 11.41
-1/64/f/a^(7/2)/(a-b)^(1/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f* x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)/((a*(-c os(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f *x+e)+1)^2*csc(f*x+e)^2+a)/((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)^2)^(1/2)/((- cos(f*x+e)+1)^2*csc(f*x+e)^2-1)/(-cos(f*x+e)+1)^4*((-cos(f*x+e)+1)^6*(a*(- cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos( f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(5/2)*(a-b)^(1/2)*csc(f*x+e)^2-11*(a*( -cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos (f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(5/2)*(-cos(f*x+e)+1)^4*(a-b)^(1/2)-6 *(a-b)^(1/2)*a^(3/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1) ^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*b*(-cos(f*x+e) +1)^4+32*ln((a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+(a*(-cos(f*x+e)+1)^4*csc(f*x +e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^ 2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*a^3*(-cos(f*x+e)+1)^4*(a-b)^(1/2)-32*ln (2/(-cos(f*x+e)+1)^2*(-a*(-cos(f*x+e)+1)^2+2*b*(-cos(f*x+e)+1)^2+(a*(-cos( f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+ e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2))*a^3*(- cos(f*x+e)+1)^4*(a-b)^(1/2)+64*ln(4*(-a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+b*( -cos(f*x+e)+1)^2*csc(f*x+e)^2+(a-b)^(1/2)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^ 4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2...
Time = 0.34 (sec) , antiderivative size = 857, normalized size of antiderivative = 5.16 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\left [\frac {8 \, \sqrt {a - b} a^{3} \log \left (\frac {b \tan \left (f x + e\right )^{2} + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{4} + {\left (8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) \tan \left (f x + e\right )^{4} - 2 \, {\left (2 \, a^{3} - 2 \, a^{2} b - {\left (4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, {\left (a^{4} - a^{3} b\right )} f \tan \left (f x + e\right )^{4}}, \frac {16 \, a^{3} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) \tan \left (f x + e\right )^{4} + {\left (8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) \tan \left (f x + e\right )^{4} - 2 \, {\left (2 \, a^{3} - 2 \, a^{2} b - {\left (4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, {\left (a^{4} - a^{3} b\right )} f \tan \left (f x + e\right )^{4}}, \frac {4 \, \sqrt {a - b} a^{3} \log \left (\frac {b \tan \left (f x + e\right )^{2} + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{4} + {\left (8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) \tan \left (f x + e\right )^{4} - {\left (2 \, a^{3} - 2 \, a^{2} b - {\left (4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, {\left (a^{4} - a^{3} b\right )} f \tan \left (f x + e\right )^{4}}, \frac {8 \, a^{3} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) \tan \left (f x + e\right )^{4} + {\left (8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) \tan \left (f x + e\right )^{4} - {\left (2 \, a^{3} - 2 \, a^{2} b - {\left (4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, {\left (a^{4} - a^{3} b\right )} f \tan \left (f x + e\right )^{4}}\right ] \]
[1/16*(8*sqrt(a - b)*a^3*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 - 2*(2*a^3 - 2* a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^4), 1/16*(16*a^3*sqrt(-a + b)*arctan(-sq rt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^3 - 4 *a^2*b - a*b^2 - 3*b^3)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 - 2*(2*a^3 - 2*a^ 2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a) )/((a^4 - a^3*b)*f*tan(f*x + e)^4), 1/8*(4*sqrt(a - b)*a^3*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e) ^2 + 1))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(-a)*arcta n(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 - (2*a^3 - 2*a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/( (a^4 - a^3*b)*f*tan(f*x + e)^4), 1/8*(8*a^3*sqrt(-a + b)*arctan(-sqrt(b*ta n(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*ta n(f*x + e)^4 - (2*a^3 - 2*a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2 )*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^4)]
\[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{5}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{5}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \]
Time = 11.69 (sec) , antiderivative size = 1215, normalized size of antiderivative = 7.32 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\text {Too large to display} \]
- (((a + b*tan(e + f*x)^2)^(1/2)*(4*a*b + 5*b^2))/(8*a) - (b*(a + b*tan(e + f*x)^2)^(3/2)*(4*a + 3*b))/(8*a^2))/(f*(a + b*tan(e + f*x)^2)^2 + a^2*f - 2*a*f*(a + b*tan(e + f*x)^2)) - (atan(((((((3*a^2*b^5*f^2)/2 + (a^3*b^4* f^2)/2 + 2*a^4*b^3*f^2)/(2*a^4*f^3) - ((a + b*tan(e + f*x)^2)^(1/2)*(256*a ^4*b^3*f^2 - 512*a^5*b^2*f^2))/(128*a^4*f^3*(a - b)^(1/2)))/(2*f*(a - b)^( 1/2)) - ((a + b*tan(e + f*x)^2)^(1/2)*(24*a*b^5 + 9*b^6 + 64*a^2*b^4 + 64* a^3*b^3 + 128*a^4*b^2))/(64*a^4*f^2))*1i)/(f*(a - b)^(1/2)) - (((((3*a^2*b ^5*f^2)/2 + (a^3*b^4*f^2)/2 + 2*a^4*b^3*f^2)/(2*a^4*f^3) + ((a + b*tan(e + f*x)^2)^(1/2)*(256*a^4*b^3*f^2 - 512*a^5*b^2*f^2))/(128*a^4*f^3*(a - b)^( 1/2)))/(2*f*(a - b)^(1/2)) + ((a + b*tan(e + f*x)^2)^(1/2)*(24*a*b^5 + 9*b ^6 + 64*a^2*b^4 + 64*a^3*b^3 + 128*a^4*b^2))/(64*a^4*f^2))*1i)/(f*(a - b)^ (1/2)))/(((((3*a^2*b^5*f^2)/2 + (a^3*b^4*f^2)/2 + 2*a^4*b^3*f^2)/(2*a^4*f^ 3) - ((a + b*tan(e + f*x)^2)^(1/2)*(256*a^4*b^3*f^2 - 512*a^5*b^2*f^2))/(1 28*a^4*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) - ((a + b*tan(e + f*x)^2)^( 1/2)*(24*a*b^5 + 9*b^6 + 64*a^2*b^4 + 64*a^3*b^3 + 128*a^4*b^2))/(64*a^4*f ^2))/(f*(a - b)^(1/2)) + ((((3*a^2*b^5*f^2)/2 + (a^3*b^4*f^2)/2 + 2*a^4*b^ 3*f^2)/(2*a^4*f^3) + ((a + b*tan(e + f*x)^2)^(1/2)*(256*a^4*b^3*f^2 - 512* a^5*b^2*f^2))/(128*a^4*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) + ((a + b*t an(e + f*x)^2)^(1/2)*(24*a*b^5 + 9*b^6 + 64*a^2*b^4 + 64*a^3*b^3 + 128*a^4 *b^2))/(64*a^4*f^2))/(f*(a - b)^(1/2)) - ((3*a*b^5)/4 + (9*b^6)/32 + (5...