\(\int \frac {\cot ^5(e+f x)}{(a+b \tan ^2(e+f x))^{5/2}} \, dx\) [351]

Optimal result

Integrand size = 25, antiderivative size = 272 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\left (8 a^2+20 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2} f}+\frac {b \left (12 a^2+15 a b-35 b^2\right )}{24 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^4(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{8 a^4 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}} \] Output:

-1/8*(8*a^2+20*a*b+35*b^2)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(9/ 
2)/f+arctanh((a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(5/2)/f+1/24*b*(1 
2*a^2+15*a*b-35*b^2)/a^3/(a-b)/f/(a+b*tan(f*x+e)^2)^(3/2)+1/8*(4*a+7*b)*co 
t(f*x+e)^2/a^2/f/(a+b*tan(f*x+e)^2)^(3/2)-1/4*cot(f*x+e)^4/a/f/(a+b*tan(f* 
x+e)^2)^(3/2)+1/8*b*(4*a^3+3*a^2*b-50*a*b^2+35*b^3)/a^4/(a-b)^2/f/(a+b*tan 
(f*x+e)^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.34 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.61 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\cot ^2(e+f x) \left (8 a^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (3 a \cot ^2(e+f x) \left (-4 a-7 b+2 a \cot ^2(e+f x)\right )-\left (8 a^2+20 a b+35 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \tan ^2(e+f x)}{a}\right )\right )\right )}{24 a^3 (-a+b) f \left (b+a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}} \] Input:

Integrate[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]
 

Output:

(Cot[e + f*x]^2*(8*a^3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f*x 
]^2)/(a - b)] + (a - b)*(3*a*Cot[e + f*x]^2*(-4*a - 7*b + 2*a*Cot[e + f*x] 
^2) - (8*a^2 + 20*a*b + 35*b^2)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Ta 
n[e + f*x]^2)/a])))/(24*a^3*(-a + b)*f*(b + a*Cot[e + f*x]^2)*Sqrt[a + b*T 
an[e + f*x]^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 4153, 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.

\(\displaystyle \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^5 \left (a+b \tan (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \frac {\int \frac {\cot ^5(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\int \frac {\cot ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {-\frac {\int \frac {\cot ^2(e+f x) \left (7 b \tan ^2(e+f x)+4 a+7 b\right )}{2 \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan ^2(e+f x)}{2 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\cot ^2(e+f x) \left (7 b \tan ^2(e+f x)+4 a+7 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan ^2(e+f x)}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\cot (e+f x) \left (8 a^2+20 b a+35 b^2+5 b (4 a+7 b) \tan ^2(e+f x)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan ^2(e+f x)}{a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\cot (e+f x) \left (8 a^2+20 b a+35 b^2+5 b (4 a+7 b) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan ^2(e+f x)}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {-\frac {-\frac {\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {2 \int -\frac {3 \cot (e+f x) \left (b \left (12 a^2+15 b a-35 b^2\right ) \tan ^2(e+f x)+(a-b) \left (8 a^2+20 b a+35 b^2\right )\right )}{2 \left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{3 a (a-b)}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {\cot (e+f x) \left (b \left (12 a^2+15 b a-35 b^2\right ) \tan ^2(e+f x)+(a-b) \left (8 a^2+20 b a+35 b^2\right )\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{a (a-b)}+\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {-\frac {-\frac {\frac {\frac {2 b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}-\frac {2 \int -\frac {\cot (e+f x) \left (\left (8 a^2+20 b a+35 b^2\right ) (a-b)^2+b \left (4 a^3+3 b a^2-50 b^2 a+35 b^3\right ) \tan ^2(e+f x)\right )}{2 \left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}}{a (a-b)}+\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\int \frac {\cot (e+f x) \left (\left (8 a^2+20 b a+35 b^2\right ) (a-b)^2+b \left (4 a^3+3 b a^2-50 b^2 a+35 b^3\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}+\frac {2 b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{a (a-b)}+\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {-\frac {-\frac {\frac {\frac {(a-b)^2 \left (8 a^2+20 a b+35 b^2\right ) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-8 a^4 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}+\frac {2 b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{a (a-b)}+\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\frac {2 (a-b)^2 \left (8 a^2+20 a b+35 b^2\right ) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {16 a^4 \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{a (a-b)}+\frac {2 b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{a (a-b)}+\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {-\frac {\frac {2 b \left (12 a^2+15 a b-35 b^2\right )}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\frac {\frac {16 a^4 \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (a-b)^2 \left (8 a^2+20 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a-b)}+\frac {2 b \left (4 a^3+3 a^2 b-50 a b^2+35 b^3\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{a (a-b)}}{2 a}-\frac {(4 a+7 b) \cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{4 a}-\frac {\cot ^2(e+f x)}{2 a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

Input:

Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]
 

Output:

(-1/2*Cot[e + f*x]^2/(a*(a + b*Tan[e + f*x]^2)^(3/2)) - (-(((4*a + 7*b)*Co 
t[e + f*x])/(a*(a + b*Tan[e + f*x]^2)^(3/2))) - ((2*b*(12*a^2 + 15*a*b - 3 
5*b^2))/(3*a*(a - b)*(a + b*Tan[e + f*x]^2)^(3/2)) + (((-2*(a - b)^2*(8*a^ 
2 + 20*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] 
+ (16*a^4*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/(a 
*(a - b)) + (2*b*(4*a^3 + 3*a^2*b - 50*a*b^2 + 35*b^3))/(a*(a - b)*Sqrt[a 
+ b*Tan[e + f*x]^2]))/(a*(a - b)))/(2*a))/(4*a))/(2*f)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 114

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])
 

rule 168

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]
 

rule 169

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]
 

rule 174

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]
 

rule 221

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]
 

rule 354

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4153

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]))
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(76696\) vs. \(2(242)=484\).

Time = 38.76 (sec) , antiderivative size = 76697, normalized size of antiderivative = 281.97

method	result	size
default	\(\text {Expression too large to display}\)	\(76697\)

Input:

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (242) = 484\).

Time = 0.21 (sec) , antiderivative size = 2411, normalized size of antiderivative = 8.86 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/48*(24*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + 
 e)^4)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sq 
rt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + 3*((8*a^5*b^2 - 4*a^4*b^3 - a 
^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a 
^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8 
*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + 
 e)^4)*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) - 2*(6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 
3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^ 
6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x 
 + e)^4 - 3*(4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x 
 + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a 
^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*t 
an(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 
-1/48*(48*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + 
 e)^4)*sqrt(-a + b)*arctan(sqrt(-a + b)/sqrt(b*tan(f*x + e)^2 + a)) - 3*(( 
8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x 
+ e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a 
*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^ 
4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(...
 

Sympy [F]

\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**5/(a+b*tan(f*x+e)**2)**(5/2),x)
 

Output:

Integral(cot(e + f*x)**5/(a + b*tan(e + f*x)**2)**(5/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 9.74 (sec) , antiderivative size = 4652, normalized size of antiderivative = 17.10 \[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

int(cot(e + f*x)^5/(a + b*tan(e + f*x)^2)^(5/2),x)
 

Output:

((b*(a + b*tan(e + f*x)^2)^2*(15*a^2*b - 250*a*b^2 + 12*a^3 + 175*b^3))/(2 
4*(a^3*b - a^4)*(a - b)) - b^3/(3*a*(a - b)) + (b*(a + b*tan(e + f*x)^2)^3 
*(3*a^2*b - 50*a*b^2 + 4*a^3 + 35*b^3))/(8*(a^3*b - a^4)*(a*b - a^2)) + (b 
*(10*a*b^2 - 7*b^3)*(a + b*tan(e + f*x)^2))/(3*a*(a - b)*(a*b - a^2)))/(f* 
(a + b*tan(e + f*x)^2)^(7/2) + a^2*f*(a + b*tan(e + f*x)^2)^(3/2) - 2*a*f* 
(a + b*tan(e + f*x)^2)^(5/2)) - (atan((a^16*f^3*(a + b*tan(e + f*x)^2)^(1/ 
2)*128i - a^11*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4 
*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*128i + b^11*f*(a + b 
*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10 
*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*1225i + a^8*b^8*f^3*(a + b*tan(e + f*x)^2)^ 
(1/2)*64i - a^9*b^7*f^3*(a + b*tan(e + f*x)^2)^(1/2)*576i + a^10*b^6*f^3*( 
a + b*tan(e + f*x)^2)^(1/2)*2240i - a^11*b^5*f^3*(a + b*tan(e + f*x)^2)^(1 
/2)*4928i + a^12*b^4*f^3*(a + b*tan(e + f*x)^2)^(1/2)*6720i - a^13*b^3*f^3 
*(a + b*tan(e + f*x)^2)^(1/2)*5824i + a^14*b^2*f^3*(a + b*tan(e + f*x)^2)^ 
(1/2)*3136i - a^15*b*f^3*(a + b*tan(e + f*x)^2)^(1/2)*960i + a^2*b^9*f*(a 
+ b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 
 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)*16885i - a^3*b^8*f*(a + b*tan(e + f*x)^2 
)^(1/2)*(a^5*f^2 - b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 
10*a^3*b^2*f^2)*19875i + a^4*b^7*f*(a + b*tan(e + f*x)^2)^(1/2)*(a^5*f^2 - 
 b^5*f^2 + 5*a*b^4*f^2 - 5*a^4*b*f^2 - 10*a^2*b^3*f^2 + 10*a^3*b^2*f^2)...
 

Reduce [F]

\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot \left (f x +e \right )^{5}}{\left (\tan \left (f x +e \right )^{2} b +a \right )^{\frac {5}{2}}}d x \] Input:

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x)
 

Output:

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x)