\(\int \frac {\cot ^4(e+f x)}{(a+b \tan ^2(e+f x))^{5/2}} \, dx\) [357]

Optimal result

Integrand size = 25, antiderivative size = 249 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {b \cot ^3(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(3 a-2 b) b \cot ^3(e+f x)}{a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^4 (a-b)^2 f}-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b)^2 f} \] Output:

arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))/(a-b)^(5/2)/f-1/3* 
b*cot(f*x+e)^3/a/(a-b)/f/(a+b*tan(f*x+e)^2)^(3/2)-(3*a-2*b)*b*cot(f*x+e)^3 
/a^2/(a-b)^2/f/(a+b*tan(f*x+e)^2)^(1/2)+1/3*(a-2*b)*(3*a^2+8*a*b-8*b^2)*co 
t(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)/a^4/(a-b)^2/f-1/3*(a^2-12*a*b+8*b^2)*cot 
(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2)/a^3/(a-b)^2/f
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 17.38 (sec) , antiderivative size = 871, normalized size of antiderivative = 3.50 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {-\frac {b \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \sqrt {1+\cos (2 (e+f x))} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{4 a \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticPi}\left (-\frac {b}{a-b},\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{(a-b)^2 f}+\frac {\sqrt {\frac {a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {4 (a \cos (e+f x)+2 b \cos (e+f x)) \csc (e+f x)}{3 a^4}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^3}+\frac {2 b^4 \sin (2 (e+f x))}{3 a^3 (a-b)^2 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}-\frac {4 \left (3 a b^3 \sin (2 (e+f x))-2 b^4 \sin (2 (e+f x))\right )}{3 a^4 (a-b)^2 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}\right )}{f} \] Input:

Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]
 

Output:

(-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sqrt 
[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + f*x]^2) 
/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e 
+ f*x)]*EllipticF[ArcSin[Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + 
f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(a*(a + b + (a - b)*Cos[2*(e + f*x 
)]))) - (4*b*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[(a + b + (a - b)*Cos[2*(e + f 
*x)])/(1 + Cos[2*(e + f*x)])]*((Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 
 + Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + 
 f*x)])*Csc[e + f*x]^2)/b]*Csc[2*(e + f*x)]*EllipticF[ArcSin[Sqrt[((a + b 
+ (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4 
)/(4*a*Sqrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]) 
- (Sqrt[-((a*Cot[e + f*x]^2)/b)]*Sqrt[-((a*(1 + Cos[2*(e + f*x)])*Csc[e + 
f*x]^2)/b)]*Sqrt[((a + b + (a - b)*Cos[2*(e + f*x)])*Csc[e + f*x]^2)/b]*Cs 
c[2*(e + f*x)]*EllipticPi[-(b/(a - b)), ArcSin[Sqrt[((a + b + (a - b)*Cos[ 
2*(e + f*x)])*Csc[e + f*x]^2)/b]/Sqrt[2]], 1]*Sin[e + f*x]^4)/(2*(a - b)*S 
qrt[1 + Cos[2*(e + f*x)]]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]])))/Sqrt[a 
 + b + (a - b)*Cos[2*(e + f*x)]])/((a - b)^2*f) + (Sqrt[(a + b + a*Cos[2*( 
e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*((4*(a*Cos[e + f*x 
] + 2*b*Cos[e + f*x])*Csc[e + f*x])/(3*a^4) - (Cot[e + f*x]*Csc[e + f*x]^2 
)/(3*a^3) + (2*b^4*Sin[2*(e + f*x)])/(3*a^3*(a - b)^2*(a + b + a*Cos[2*...
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4153, 374, 27, 441, 445, 445, 27, 291, 216}

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.

\(\displaystyle \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \left (a+b \tan (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 374

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 441

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {-\frac {\left (a^2-12 a b+8 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a}-\frac {-\frac {3 a^3 \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {a-b}}-\frac {(a-2 b) \left (3 a^2+8 a b-8 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{3 a}}{a (a-b)}-\frac {b (3 a-2 b) \cot ^3(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{a (a-b)}-\frac {b \cot ^3(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\)

Input:

Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]
 

Output:

(-1/3*(b*Cot[e + f*x]^3)/(a*(a - b)*(a + b*Tan[e + f*x]^2)^(3/2)) + (-(((3 
*a - 2*b)*b*Cot[e + f*x]^3)/(a*(a - b)*Sqrt[a + b*Tan[e + f*x]^2])) + (-1/ 
3*((a^2 - 12*a*b + 8*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/a - ( 
(-3*a^3*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/Sqr 
t[a - b] - ((a - 2*b)*(3*a^2 + 8*a*b - 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[ 
e + f*x]^2])/a)/(3*a))/(a*(a - b)))/(a*(a - b)))/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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 441

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
+ b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si 
mp[1/(a*2*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 
)^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m 
 + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, 
 x] && LtQ[p, -1]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

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 [F]

\[\int \frac {\cot \left (f x +e \right )^{4}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}d x\]

Input:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x)
 

Output:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 879, normalized size of antiderivative = 3.53 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[-1/12*(3*(a^4*b^2*tan(f*x + e)^7 + 2*a^5*b*tan(f*x + e)^5 + a^6*tan(f*x + 
 e)^3)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 
- 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + 
e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + 
 e)^2 + 1)) - 4*((3*a^4*b^2 - a^3*b^3 - 26*a^2*b^4 + 40*a*b^5 - 16*b^6)*ta 
n(f*x + e)^6 - a^6 + 3*a^5*b - 3*a^4*b^2 + a^3*b^3 + 3*(2*a^5*b - a^4*b^2 
- 13*a^3*b^3 + 20*a^2*b^4 - 8*a*b^5)*tan(f*x + e)^4 + 3*(a^6 - a^5*b - 3*a 
^4*b^2 + 5*a^3*b^3 - 2*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)^7 + 2*(a^8*b 
 - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3* 
a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^3), 1/6*(3*(a^4*b^2*tan(f*x + e)^7 + 2*a 
^5*b*tan(f*x + e)^5 + a^6*tan(f*x + e)^3)*sqrt(a - b)*arctan(-2*sqrt(b*tan 
(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) 
+ 2*((3*a^4*b^2 - a^3*b^3 - 26*a^2*b^4 + 40*a*b^5 - 16*b^6)*tan(f*x + e)^6 
 - a^6 + 3*a^5*b - 3*a^4*b^2 + a^3*b^3 + 3*(2*a^5*b - a^4*b^2 - 13*a^3*b^3 
 + 20*a^2*b^4 - 8*a*b^5)*tan(f*x + e)^4 + 3*(a^6 - a^5*b - 3*a^4*b^2 + 5*a 
^3*b^3 - 2*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)^7 + 2*(a^8*b - 3*a^7*b^2 
 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^ 
6*b^3)*f*tan(f*x + e)^3)]
 

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**4/(a+b*tan(f*x+e)**2)**(5/2),x)
 

Output:

Integral(cot(e + f*x)**4/(a + b*tan(e + f*x)**2)**(5/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \] Input:

int(cot(e + f*x)^4/(a + b*tan(e + f*x)^2)^(5/2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot \left (f x +e \right )^{4}}{\left (\tan \left (f x +e \right )^{2} b +a \right )^{\frac {5}{2}}}d x \] Input:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x)
 

Output:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x)