\(\int \frac {\cot ^3(e+f x)}{(a+b \tan ^2(e+f x))^{5/2}} \, dx\) [350]

Optimal result

Integrand size = 25, antiderivative size = 206 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {(2 a+5 b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{7/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2} f}-\frac {(3 a-5 b) b}{6 a^2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^2(e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {b \left (a^2-8 a b+5 b^2\right )}{2 a^3 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}} \] Output:

1/2*(2*a+5*b)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(7/2)/f-arctanh( 
(a+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(5/2)/f-1/6*(3*a-5*b)*b/a^2/(a 
-b)/f/(a+b*tan(f*x+e)^2)^(3/2)-1/2*cot(f*x+e)^2/a/f/(a+b*tan(f*x+e)^2)^(3/ 
2)-1/2*b*(a^2-8*a*b+5*b^2)/a^3/(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 = 0.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\cot ^2(e+f x) \left (-2 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \left (3 a \cot ^2(e+f x)+(2 a+5 b) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \tan ^2(e+f x)}{a}\right )\right )\right )}{6 a^2 (-a+b) f \left (b+a \cot ^2(e+f x)\right ) \sqrt {a+b \tan ^2(e+f x)}} \] Input:

Integrate[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]
 

Output:

(Cot[e + f*x]^2*(-2*a^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[e + f* 
x]^2)/(a - b)] + (a - b)*(3*a*Cot[e + f*x]^2 + (2*a + 5*b)*Hypergeometric2 
F1[-3/2, 1, -1/2, 1 + (b*Tan[e + f*x]^2)/a])))/(6*a^2*(-a + b)*f*(b + a*Co 
t[e + f*x]^2)*Sqrt[a + b*Tan[e + f*x]^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.43 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4153, 354, 114, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {4 a^3 \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (a-b)^2 (2 a+5 b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a-b)}+\frac {2 b \left (a^2-8 a b+5 b^2\right )}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{a (a-b)}+\frac {2 b (3 a-5 b)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 a}-\frac {\cot (e+f x)}{a \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{2 f}\)

Input:

Int[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]
 

Output:

(-(Cot[e + f*x]/(a*(a + b*Tan[e + f*x]^2)^(3/2))) - ((2*(3*a - 5*b)*b)/(3* 
a*(a - b)*(a + b*Tan[e + f*x]^2)^(3/2)) + (((-2*(a - b)^2*(2*a + 5*b)*ArcT 
anh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (4*a^3*ArcTanh[Sqrt[a + 
 b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/(a*(a - b)) + (2*b*(a^2 - 8* 
a*b + 5*b^2))/(a*(a - b)*Sqrt[a + b*Tan[e + f*x]^2]))/(a*(a - b)))/(2*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 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. \(56969\) vs. \(2(180)=360\).

Time = 22.03 (sec) , antiderivative size = 56970, normalized size of antiderivative = 276.55

method	result	size
default	\(\text {Expression too large to display}\)	\(56970\)

Input:

int(cot(f*x+e)^3/(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. 489 vs. \(2 (180) = 360\).

Time = 0.22 (sec) , antiderivative size = 2061, normalized size of antiderivative = 10.00 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(6*(a^4*b^2*tan(f*x + e)^6 + 2*a^5*b*tan(f*x + e)^4 + a^6*tan(f*x + 
e)^2)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqr 
t(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + 3*((2*a^4*b^2 - a^3*b^3 - 9*a^ 
2*b^4 + 13*a*b^5 - 5*b^6)*tan(f*x + e)^6 + 2*(2*a^5*b - a^4*b^2 - 9*a^3*b^ 
3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + (2*a^6 - a^5*b - 9*a^4*b^2 + 13 
*a^3*b^3 - 5*a^2*b^4)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sq 
rt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(3*a^6 - 9*a^5 
*b + 9*a^4*b^2 - 3*a^3*b^3 + 3*(a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5 
)*tan(f*x + e)^4 + 2*(3*a^5*b - 19*a^4*b^2 + 26*a^3*b^3 - 10*a^2*b^4)*tan( 
f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 
- a^4*b^5)*f*tan(f*x + e)^6 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)* 
f*tan(f*x + e)^4 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^2) 
, 1/12*(12*(a^4*b^2*tan(f*x + e)^6 + 2*a^5*b*tan(f*x + e)^4 + a^6*tan(f*x 
+ e)^2)*sqrt(-a + b)*arctan(sqrt(-a + b)/sqrt(b*tan(f*x + e)^2 + a)) + 3*( 
(2*a^4*b^2 - a^3*b^3 - 9*a^2*b^4 + 13*a*b^5 - 5*b^6)*tan(f*x + e)^6 + 2*(2 
*a^5*b - a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + (2*a 
^6 - a^5*b - 9*a^4*b^2 + 13*a^3*b^3 - 5*a^2*b^4)*tan(f*x + e)^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) - 2*(3*a^6 - 9*a^5*b + 9*a^4*b^2 - 3*a^3*b^3 + 3*(a^4*b^2 - 9*a^3 
*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + 2*(3*a^5*b - 19*a^4*b^2 +...
 

Sympy [F]

\[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{3}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**3/(a+b*tan(f*x+e)**2)**(5/2),x)
 

Output:

Integral(cot(e + f*x)**3/(a + b*tan(e + f*x)**2)**(5/2), x)
                                                                                    
                                                                                    
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 9.08 (sec) , antiderivative size = 3429, normalized size of antiderivative = 16.65 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

int(cot(e + f*x)^3/(a + b*tan(e + f*x)^2)^(5/2),x)
 

Output:

(atan(((((a + b*tan(e + f*x)^2)^(1/2)*(400*a^9*b^14*f^3 - 3680*a^10*b^13*f 
^3 + 14864*a^11*b^12*f^3 - 34240*a^12*b^11*f^3 + 48480*a^13*b^10*f^3 - 412 
80*a^14*b^9*f^3 + 16864*a^15*b^8*f^3 + 2688*a^16*b^7*f^3 - 6000*a^17*b^6*f 
^3 + 1440*a^18*b^5*f^3 + 1040*a^19*b^4*f^3 - 704*a^20*b^3*f^3 + 128*a^21*b 
^2*f^3) - ((2*a + 5*b)*(320*a^12*b^14*f^4 - 3392*a^13*b^13*f^4 + 16192*a^1 
4*b^12*f^4 - 45760*a^15*b^11*f^4 + 84608*a^16*b^10*f^4 - 106624*a^17*b^9*f 
^4 + 92288*a^18*b^8*f^4 - 53632*a^19*b^7*f^4 + 19520*a^20*b^6*f^4 - 3648*a 
^21*b^5*f^4 + 64*a^22*b^4*f^4 + 64*a^23*b^3*f^4 - ((a + b*tan(e + f*x)^2)^ 
(1/2)*(2*a + 5*b)*(256*a^15*b^13*f^5 - 3072*a^16*b^12*f^5 + 16640*a^17*b^1 
1*f^5 - 53760*a^18*b^10*f^5 + 115200*a^19*b^9*f^5 - 172032*a^20*b^8*f^5 + 
182784*a^21*b^7*f^5 - 138240*a^22*b^6*f^5 + 72960*a^23*b^5*f^5 - 25600*a^2 
4*b^4*f^5 + 5376*a^25*b^3*f^5 - 512*a^26*b^2*f^5))/(4*f*(a^7)^(1/2))))/(4* 
f*(a^7)^(1/2)))*(2*a + 5*b)*1i)/(4*f*(a^7)^(1/2)) + (((a + b*tan(e + f*x)^ 
2)^(1/2)*(400*a^9*b^14*f^3 - 3680*a^10*b^13*f^3 + 14864*a^11*b^12*f^3 - 34 
240*a^12*b^11*f^3 + 48480*a^13*b^10*f^3 - 41280*a^14*b^9*f^3 + 16864*a^15* 
b^8*f^3 + 2688*a^16*b^7*f^3 - 6000*a^17*b^6*f^3 + 1440*a^18*b^5*f^3 + 1040 
*a^19*b^4*f^3 - 704*a^20*b^3*f^3 + 128*a^21*b^2*f^3) + ((2*a + 5*b)*(320*a 
^12*b^14*f^4 - 3392*a^13*b^13*f^4 + 16192*a^14*b^12*f^4 - 45760*a^15*b^11* 
f^4 + 84608*a^16*b^10*f^4 - 106624*a^17*b^9*f^4 + 92288*a^18*b^8*f^4 - 536 
32*a^19*b^7*f^4 + 19520*a^20*b^6*f^4 - 3648*a^21*b^5*f^4 + 64*a^22*b^4*...
 

Reduce [F]

\[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot \left (f x +e \right )^{3}}{\left (\tan \left (f x +e \right )^{2} b +a \right )^{\frac {5}{2}}}d x \] Input:

int(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x)
 

Output:

int(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x)