Integrand size = 25, antiderivative size = 161 \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (8 a^2-12 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {(4 a-5 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 f}-\frac {a \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 f} \]
(a-b)^(3/2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/f-1/8*(8*a^2-12* a*b+3*b^2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/f/a^(1/2)+1/8*(4*a-5* b)*cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2)/f-1/4*a*cot(f*x+e)^4*(a+b*tan(f*x +e)^2)^(1/2)/f
Time = 1.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.87 \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\left (-8 a^2+12 a b-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 (a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+\cot ^2(e+f x) \left (4 a-5 b-2 a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}\right )}{8 \sqrt {a} f} \]
((-8*a^2 + 12*a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + S qrt[a]*(8*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + Cot[e + f*x]^2*(4*a - 5*b - 2*a*Cot[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2] ))/(8*Sqrt[a]*f)
Time = 0.40 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.99, 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, 109, 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.
| \(\displaystyle \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \tan (e+f x)^2\right )^{3/2}}{\tan (e+f x)^5}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^5(e+f x) \left (b \tan ^2(e+f x)+a\right )^{3/2}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot ^3(e+f x) \left (b \tan ^2(e+f x)+a\right )^{3/2}}{\tan ^2(e+f x)+1}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {-\frac {1}{2} \int \frac {\cot ^2(e+f x) \left ((3 a-4 b) b \tan ^2(e+f x)+a (4 a-5 b)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{4} \int \frac {\cot ^2(e+f x) \left ((3 a-4 b) b \tan ^2(e+f x)+a (4 a-5 b)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {\int \frac {a \cot (e+f x) \left (8 a^2-12 b a+3 b^2+(4 a-5 b) b \tan ^2(e+f x)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a}+(4 a-5 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \int \frac {\cot (e+f x) \left (8 a^2-12 b a+3 b^2+(4 a-5 b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)+(4 a-5 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (\left (8 a^2-12 a b+3 b^2\right ) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-8 (a-b)^2 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)\right )+(4 a-5 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (8 a^2-12 a b+3 b^2\right ) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {16 (a-b)^2 \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}\right )+(4 a-5 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (16 (a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )-\frac {2 \left (8 a^2-12 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}\right )+(4 a-5 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )-\frac {1}{2} a \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}\) |
(-1/2*(a*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2]) + (((-2*(8*a^2 - 12*a* b + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + 16*(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/2 + (4*a - 5*b)* Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/4)/(2*f)
3.4.11.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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. \(2241\) vs. \(2(139)=278\).
Time = 0.91 (sec) , antiderivative size = 2242, normalized size of antiderivative = 13.93
-1/64/f/(a-b)^(1/2)/a^(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)/((-cos(f*x+e )+1)^2*csc(f*x+e)^2-1)^2)^(3/2)*((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)^3/(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)^(3/2)/(-cos(f*x+e)+1)^4*(a^(3/2)*(-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-b)^(1/2)*csc(f*x+e)^2-32*a^ 2*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 -b)^(1/2)*(-cos(f*x+e)+1)^4+32*a^2*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*(-c os(f*x+e)+1)^2*csc(f*x+e)^2+a)^(1/2)*a^(1/2)-a+2*b)/a^(1/2))*(a-b)^(1/2)*( -cos(f*x+e)+1)^4+48*a*ln(2/(-cos(f*x+e)+1)^2*(-a*(-cos(f*x+e)+1)^2+2*b*(-c os(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))*b*(a-b)^(1/2)*(-cos(f*x+e)+1)^4-48*a*b*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-b)^(1/2)*(-cos(f*x+e)+1)^4-11*a^(3/2)*(a*(-cos(f*x+e)+1)^4*...
Time = 0.32 (sec) , antiderivative size = 748, normalized size of antiderivative = 4.65 \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\left [-\frac {8 \, {\left (a^{2} - a b\right )} \sqrt {a - b} \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^{2} - 12 \, a b + 3 \, b^{2}\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 ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, a f \tan \left (f x + e\right )^{4}}, \frac {16 \, {\left (a^{2} - a b\right )} \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^{2} - 12 \, a b + 3 \, b^{2}\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 ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{16 \, a f \tan \left (f x + e\right )^{4}}, \frac {{\left (8 \, a^{2} - 12 \, a b + 3 \, b^{2}\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} - 4 \, {\left (a^{2} - a b\right )} \sqrt {a - b} \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 ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, a f \tan \left (f x + e\right )^{4}}, \frac {{\left (8 \, a^{2} - 12 \, a b + 3 \, b^{2}\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} + 8 \, {\left (a^{2} - a b\right )} \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 ({\left (4 \, a^{2} - 5 \, a b\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{8 \, a f \tan \left (f x + e\right )^{4}}\right ] \]
[-1/16*(8*(a^2 - a*b)*sqrt(a - b)*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^2 - 12*a*b + 3*b^2)*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*((4*a^2 - 5 *a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e )^4), 1/16*(16*(a^2 - a*b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a) *sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^2 - 12*a*b + 3*b^2)*sqrt(a)*l og((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*((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt( b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/8*((8*a^2 - 12*a*b + 3*b^2) *sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 - 4 *(a^2 - a*b)*sqrt(a - b)*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 + ((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/8*((8*a^2 - 12*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e) ^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 + 8*(a^2 - a*b)*sqrt(-a + b)*arctan(-sq rt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + ((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4)]
\[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cot ^{5}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{5} \,d x } \]
Timed out. \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
Time = 11.41 (sec) , antiderivative size = 578, normalized size of antiderivative = 3.59 \[ \int \cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (\frac {3\,a\,b^2}{8}-\frac {a^2\,b}{2}\right )+\frac {b\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}\,\left (4\,a-5\,b\right )}{8}}{f\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^2+a^2\,f-2\,a\,f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}-\frac {\mathrm {atanh}\left (\frac {9\,b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{32\,\left (\frac {a^3\,b^5}{4}-\frac {25\,a^2\,b^6}{32}+\frac {13\,a\,b^7}{16}-\frac {9\,b^8}{32}\right )}-\frac {a\,b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {a^3-3\,a^2\,b+3\,a\,b^2-b^3}}{4\,\left (\frac {a^3\,b^5}{4}-\frac {25\,a^2\,b^6}{32}+\frac {13\,a\,b^7}{16}-\frac {9\,b^8}{32}\right )}\right )\,\sqrt {{\left (a-b\right )}^3}}{f}-\frac {\mathrm {atanh}\left (\frac {75\,\sqrt {a}\,b^7\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{64\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}-\frac {159\,b^8\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{256\,\sqrt {a}\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}-\frac {29\,a^{3/2}\,b^6\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{32\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}+\frac {a^{5/2}\,b^5\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{4\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}+\frac {27\,b^9\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{256\,a^{3/2}\,\left (\frac {75\,a\,b^7}{64}-\frac {159\,b^8}{256}-\frac {29\,a^2\,b^6}{32}+\frac {a^3\,b^5}{4}+\frac {27\,b^9}{256\,a}\right )}\right )\,\left (8\,a^2-12\,a\,b+3\,b^2\right )}{8\,\sqrt {a}\,f} \]
((a + b*tan(e + f*x)^2)^(1/2)*((3*a*b^2)/8 - (a^2*b)/2) + (b*(a + b*tan(e + f*x)^2)^(3/2)*(4*a - 5*b))/8)/(f*(a + b*tan(e + f*x)^2)^2 + a^2*f - 2*a* f*(a + b*tan(e + f*x)^2)) - (atanh((9*b^6*(a + b*tan(e + f*x)^2)^(1/2)*(3* a*b^2 - 3*a^2*b + a^3 - b^3)^(1/2))/(32*((13*a*b^7)/16 - (9*b^8)/32 - (25* a^2*b^6)/32 + (a^3*b^5)/4)) - (a*b^5*(a + b*tan(e + f*x)^2)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)^(1/2))/(4*((13*a*b^7)/16 - (9*b^8)/32 - (25*a^2*b^ 6)/32 + (a^3*b^5)/4)))*((a - b)^3)^(1/2))/f - (atanh((75*a^(1/2)*b^7*(a + b*tan(e + f*x)^2)^(1/2))/(64*((75*a*b^7)/64 - (159*b^8)/256 - (29*a^2*b^6) /32 + (a^3*b^5)/4 + (27*b^9)/(256*a))) - (159*b^8*(a + b*tan(e + f*x)^2)^( 1/2))/(256*a^(1/2)*((75*a*b^7)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3 *b^5)/4 + (27*b^9)/(256*a))) - (29*a^(3/2)*b^6*(a + b*tan(e + f*x)^2)^(1/2 ))/(32*((75*a*b^7)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5)/4 + (2 7*b^9)/(256*a))) + (a^(5/2)*b^5*(a + b*tan(e + f*x)^2)^(1/2))/(4*((75*a*b^ 7)/64 - (159*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5)/4 + (27*b^9)/(256*a))) + (27*b^9*(a + b*tan(e + f*x)^2)^(1/2))/(256*a^(3/2)*((75*a*b^7)/64 - (15 9*b^8)/256 - (29*a^2*b^6)/32 + (a^3*b^5)/4 + (27*b^9)/(256*a))))*(8*a^2 - 12*a*b + 3*b^2))/(8*a^(1/2)*f)