\(\int \frac {\cot ^4(e+f x)}{(a+b \tan ^2(e+f x))^{3/2}} \, dx\) [344]

Optimal result

Integrand size = 25, antiderivative size = 184 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{3/2} f}-\frac {b \cot ^3(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(3 a-4 b) (a+2 b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^3 (a-b) f}-\frac {(a-4 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a^2 (a-b) f} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 11.79 (sec) , antiderivative size = 1398, normalized size of antiderivative = 7.60 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]
 

Output:

-1/45*(Cos[e + f*x]^2*Cot[e + f*x]^3*((45*a*Csc[e + f*x]^2)/(a - b) - (270 
*b*Sec[e + f*x]^2)/(a - b) + (4*(a - b)*Hypergeometric2F1[2, 2, 7/2, ((a - 
 b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/a - (24*(a - b)*HypergeometricPFQ[{ 
2, 2, 2}, {1, 7/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2)/a - (16*(a 
 - b)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 7/2}, ((a - b)*Sin[e + f*x]^2 
)/a]*Sin[e + f*x]^2)/a - (1080*b^2*Sec[e + f*x]^2*Tan[e + f*x]^2)/(a*(a - 
b)) - (132*(a - b)*b*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2) 
/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/a^2 - (144*(a - b)*b*HypergeometricPFQ[ 
{2, 2, 2}, {1, 7/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f* 
x]^2)/a^2 - (48*(a - b)*b*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 7/2}, ((a 
 - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^2)/a^2 - (720*b^3*Sec 
[e + f*x]^2*Tan[e + f*x]^4)/(a^2*(a - b)) - (312*(a - b)*b^2*Hypergeometri 
c2F1[2, 2, 7/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4) 
/a^3 - (216*(a - b)*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Si 
n[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^4)/a^3 - (48*(a - b)*b^2*Hype 
rgeometricPFQ[{2, 2, 2, 2}, {1, 1, 7/2}, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e 
 + f*x]^2*Tan[e + f*x]^4)/a^3 - (176*(a - b)*b^3*Hypergeometric2F1[2, 2, 7 
/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*Tan[e + f*x]^6)/a^4 - (96*( 
a - b)*b^3*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Sin[e + f*x]^2) 
/a]*Sin[e + f*x]^2*Tan[e + f*x]^6)/a^4 - (16*(a - b)*b^3*Hypergeometric...
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4153, 374, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 374

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 216

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

Input:

Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]
 

Output:

(-((b*Cot[e + f*x]^3)/(a*(a - b)*Sqrt[a + b*Tan[e + f*x]^2])) + (-1/3*((a 
- 4*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/a - ((-3*a^2*ArcTan[(Sqr 
t[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/Sqrt[a - b] - ((3*a - 
4*b)*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/(3*a))/(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 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 [C] (warning: unable to verify)

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

Time = 14.81 (sec) , antiderivative size = 1080, normalized size of antiderivative = 5.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1080\)

Input:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/24/f/a^3/((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)/(a-b)*(-((2*I*b^(1/2) 
*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^2*b+((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/ 
2)*a^3+((1-cos(f*x+e))^8*csc(f*x+e)^8-16*(1-cos(f*x+e))^6*csc(f*x+e)^6+30* 
(1-cos(f*x+e))^4*csc(f*x+e)^4-16*(1-cos(f*x+e))^2*csc(f*x+e)^2)*a^3*((2*I* 
b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)+128*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a 
)^(1/2)*b^3*(1-cos(f*x+e))^4*csc(f*x+e)^4+(-(cos(f*x+e)-1)^8*csc(f*x+e)^8- 
46*(1-cos(f*x+e))^4*csc(f*x+e)^4)*b*a^2*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a 
)^(1/2)+(16*(1-cos(f*x+e))^6*csc(f*x+e)^6-64*(1-cos(f*x+e))^4*csc(f*x+e)^4 
+16*(1-cos(f*x+e))^2*csc(f*x+e)^2)*b^2*a*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/ 
a)^(1/2)+96*a^3*(1/a*(I*cos(f*x+e)*b^(1/2)*(a-b)^(1/2)-I*b^(1/2)*(a-b)^(1/ 
2)+a*cos(f*x+e)-cos(f*x+e)*b+b)/(cos(f*x+e)+1))^(1/2)*((-I*cos(f*x+e)*b^(1 
/2)*(a-b)^(1/2)+I*b^(1/2)*(a-b)^(1/2)+a*cos(f*x+e)-cos(f*x+e)*b+b)/a/(cos( 
f*x+e)+1))^(1/2)*EllipticF(((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*(csc( 
f*x+e)-cot(f*x+e)),((8*I*b^(3/2)*(a-b)^(1/2)-4*I*b^(1/2)*(a-b)^(1/2)*a+a^2 
-8*a*b+8*b^2)/a^2)^(1/2))*(1-cos(f*x+e))^3*csc(f*x+e)^3-192*(1/a*(I*cos(f* 
x+e)*b^(1/2)*(a-b)^(1/2)-I*b^(1/2)*(a-b)^(1/2)+a*cos(f*x+e)-cos(f*x+e)*b+b 
)/(cos(f*x+e)+1))^(1/2)*((-I*cos(f*x+e)*b^(1/2)*(a-b)^(1/2)+I*b^(1/2)*(a-b 
)^(1/2)+a*cos(f*x+e)-cos(f*x+e)*b+b)/a/(cos(f*x+e)+1))^(1/2)*EllipticPi((( 
2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*(csc(f*x+e)-cot(f*x+e)),1/(-2*I*b^ 
(1/2)*(a-b)^(1/2)-a+2*b)*a,((-2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)/(...
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.15 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a^{3} b \tan \left (f x + e\right )^{5} + a^{4} \tan \left (f x + e\right )^{3}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, {\left ({\left (3 \, a^{3} b - a^{2} b^{2} - 10 \, a b^{3} + 8 \, b^{4}\right )} \tan \left (f x + e\right )^{4} - a^{4} + 2 \, a^{3} b - a^{2} b^{2} + {\left (3 \, a^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, {\left ({\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f \tan \left (f x + e\right )^{5} + {\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \tan \left (f x + e\right )^{3}\right )}}, \frac {3 \, {\left (a^{3} b \tan \left (f x + e\right )^{5} + a^{4} \tan \left (f x + e\right )^{3}\right )} \sqrt {a - b} \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) + 2 \, {\left ({\left (3 \, a^{3} b - a^{2} b^{2} - 10 \, a b^{3} + 8 \, b^{4}\right )} \tan \left (f x + e\right )^{4} - a^{4} + 2 \, a^{3} b - a^{2} b^{2} + {\left (3 \, a^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, {\left ({\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f \tan \left (f x + e\right )^{5} + {\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \tan \left (f x + e\right )^{3}\right )}}\right ] \] Input:

integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/12*(3*(a^3*b*tan(f*x + e)^5 + a^4*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^3*b - 
a^2*b^2 - 10*a*b^3 + 8*b^4)*tan(f*x + e)^4 - a^4 + 2*a^3*b - a^2*b^2 + (3* 
a^4 - 2*a^3*b - 5*a^2*b^2 + 4*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 
 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^5 + (a^6 - 2*a^5*b + 
a^4*b^2)*f*tan(f*x + e)^3), 1/6*(3*(a^3*b*tan(f*x + e)^5 + a^4*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^3*b - a^2*b^2 - 10*a*b^3 + 
8*b^4)*tan(f*x + e)^4 - a^4 + 2*a^3*b - a^2*b^2 + (3*a^4 - 2*a^3*b - 5*a^2 
*b^2 + 4*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^ 
4*b^2 + a^3*b^3)*f*tan(f*x + e)^5 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + 
e)^3)]
 

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**4/(a+b*tan(f*x+e)**2)**(3/2),x)
 

Output:

Integral(cot(e + f*x)**4/(a + b*tan(e + f*x)**2)**(3/2), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \] Input:

int(cot(e + f*x)^4/(a + b*tan(e + f*x)^2)^(3/2),x)
 

Output:

int(cot(e + f*x)^4/(a + b*tan(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot \left (f x +e \right )^{4}}{\left (\tan \left (f x +e \right )^{2} b +a \right )^{\frac {3}{2}}}d x \] Input:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x)
 

Output:

int(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x)