\(\int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx\) [325]

Optimal result

Integrand size = 25, antiderivative size = 166 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\left (8 a^2+4 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f}+\frac {(4 a+3 b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{8 a^2 f}-\frac {\cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{4 a f} \] Output:

-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))/(a-b)^(1/2)/f+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
 

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\left (-8 a^3+4 a^2 b+a b^2+3 b^3\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 a^2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+(-a+b) \cot ^2(e+f x) \left (-4 a-3 b+2 a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}\right )}{8 a^{5/2} (a-b) f} \] Input:

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

Output:

((-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)
 

Rubi [A] (warning: unable to verify)

Time = 0.65 (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

\(\displaystyle \frac {-\frac {-\frac {\frac {16 a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 \left (8 a^2+4 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}-\frac {(4 a+3 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{4 a}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a}}{2 f}\)

Input:

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

Output:

(-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)
 

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 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. \(1234\) vs. \(2(144)=288\).

Time = 6.63 (sec) , antiderivative size = 1235, normalized size of antiderivative = 7.44

method	result	size
default	\(\text {Expression too large to display}\)	\(1235\)

Input:

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/16/f/a^(7/2)/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1 
)^2)^(1/2)/(a+b*tan(f*x+e)^2)^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/(1-c 
os(f*x+e))^4*(-16*ln(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+2*(1-cos(f*x+ 
e))^2*b+2*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2) 
*sin(f*x+e)^2+a*sin(f*x+e)^2))*a^3*(1-cos(f*x+e))^4*(a-b)^(1/2)+16*ln(2/a^ 
(1/2)*(a^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*co 
s(f*x+e)+((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)- 
a*cos(f*x+e)+cos(f*x+e)*b+b)/(cos(f*x+e)+1))*a^3*(1-cos(f*x+e))^4*(a-b)^(1 
/2)+32*ln(4*(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2) 
^(1/2)*cos(f*x+e)+4*(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+ 
e)+1)^2)^(1/2)+4*a*cos(f*x+e)-4*cos(f*x+e)*b)*a^(7/2)*(1-cos(f*x+e))^4-8*l 
n(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+2*(1-cos(f*x+e))^2*b+2*((a*cos(f 
*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin 
(f*x+e)^2))*a^2*(1-cos(f*x+e))^4*(a-b)^(1/2)*b-6*(a-b)^(1/2)*ln(2/(1-cos(f 
*x+e))^2*(-a*(1-cos(f*x+e))^2+2*(1-cos(f*x+e))^2*b+2*((a*cos(f*x+e)^2+b*si 
n(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2))* 
a*b^2*(1-cos(f*x+e))^4+8*ln(2/a^(1/2)*(a^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+ 
e)^2)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/ 
(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+cos(f*x+e)*b+b)/(cos(f*x+e)+1 
))*a^2*(1-cos(f*x+e))^4*(a-b)^(1/2)*b+6*(a-b)^(1/2)*ln(2/a^(1/2)*(a^(1/...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 836, normalized size of antiderivative = 5.04 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx =\text {Too large to display} \] Input:

integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

[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(-a + b)/sqrt(b*tan(f*x + e)^2 + a))*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)*arctan(sqrt(- 
a)/sqrt(b*tan(f*x + e)^2 + 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(-a + b)/sqrt(b 
*tan(f*x + e)^2 + a))*tan(f*x + e)^4 - (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*s 
qrt(-a)*arctan(sqrt(-a)/sqrt(b*tan(f*x + e)^2 + 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)]
 

Sympy [F]

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

integrate(cot(f*x+e)**5/(a+b*tan(f*x+e)**2)**(1/2),x)
 

Output:

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

Maxima [F(-1)]

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

integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [B] (verification not implemented)

Time = 8.33 (sec) , antiderivative size = 1215, normalized size of antiderivative = 7.32 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\text {Too large to display} \] Input:

int(cot(e + f*x)^5/(a + b*tan(e + f*x)^2)^(1/2),x)
 

Output:

- (((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...
 

Reduce [F]

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

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x)
 

Output:

int(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x)