\(\int \frac {\cot ^4(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\) [374]

Optimal result

Integrand size = 23, antiderivative size = 230 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} f}+\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{8 a^2 (a+b)^4 f}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{24 a^2 (a+b)^3 f}-\frac {b \cot ^3(e+f x)}{4 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{8 a^2 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )} \] Output:

x/a^3-1/8*b^(5/2)*(63*a^2+36*a*b+8*b^2)*arctan(b^(1/2)*tan(f*x+e)/(a+b)^(1 
/2))/a^3/(a+b)^(9/2)/f+1/8*(8*a^3+32*a^2*b-15*a*b^2-4*b^3)*cot(f*x+e)/a^2/ 
(a+b)^4/f-1/24*(8*a^2-39*a*b-12*b^2)*cot(f*x+e)^3/a^2/(a+b)^3/f-1/4*b*cot( 
f*x+e)^3/a/(a+b)/f/(a+b+b*tan(f*x+e)^2)^2-1/8*b*(11*a+4*b)*cot(f*x+e)^3/a^ 
2/(a+b)^2/f/(a+b+b*tan(f*x+e)^2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.37 (sec) , antiderivative size = 3340, normalized size of antiderivative = 14.52 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6 
*((b^3*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[ 
4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*( 
-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(64*a^3*Sqrt[a 
 + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/64)*b^3*ArcTan[Sec[f*x]*(Co 
s[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/ 
(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f* 
x] + a*Sin[2*e + f*x])]*Sin[2*e])/(a^3*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b 
*Sin[4*e]])))/((a + b)^4*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e 
 + 2*f*x])*Csc[e]*Csc[e + f*x]^3*Sec[2*e]*Sec[e + f*x]^6*(-36*a^6*f*x*Cos[ 
f*x] - 336*a^5*b*f*x*Cos[f*x] - 1560*a^4*b^2*f*x*Cos[f*x] - 3600*a^3*b^3*f 
*x*Cos[f*x] - 4260*a^2*b^4*f*x*Cos[f*x] - 2496*a*b^5*f*x*Cos[f*x] - 576*b^ 
6*f*x*Cos[f*x] + 36*a^6*f*x*Cos[3*f*x] + 240*a^5*b*f*x*Cos[3*f*x] + 408*a^ 
4*b^2*f*x*Cos[3*f*x] - 48*a^3*b^3*f*x*Cos[3*f*x] - 732*a^2*b^4*f*x*Cos[3*f 
*x] - 672*a*b^5*f*x*Cos[3*f*x] - 192*b^6*f*x*Cos[3*f*x] + 36*a^6*f*x*Cos[2 
*e - f*x] + 336*a^5*b*f*x*Cos[2*e - f*x] + 1560*a^4*b^2*f*x*Cos[2*e - f*x] 
 + 3600*a^3*b^3*f*x*Cos[2*e - f*x] + 4260*a^2*b^4*f*x*Cos[2*e - f*x] + 249 
6*a*b^5*f*x*Cos[2*e - f*x] + 576*b^6*f*x*Cos[2*e - f*x] + 36*a^6*f*x*Cos[2 
*e + f*x] + 336*a^5*b*f*x*Cos[2*e + f*x] + 1560*a^4*b^2*f*x*Cos[2*e + f*x] 
 + 3600*a^3*b^3*f*x*Cos[2*e + f*x] + 4260*a^2*b^4*f*x*Cos[2*e + f*x] + ...
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4629, 2075, 374, 441, 445, 27, 445, 397, 216, 218}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4629

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

\(\Big \downarrow \) 2075

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

\(\Big \downarrow \) 374

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

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {\frac {\int \frac {\cot ^4(e+f x) \left (8 a^2-39 b a-12 b^2-5 b (11 a+4 b) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {3 \cot ^2(e+f x) \left (8 a^3+32 b a^2-15 b^2 a-4 b^3+b \left (8 a^2-39 b a-12 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{3 (a+b)}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\cot ^2(e+f x) \left (8 a^3+32 b a^2-15 b^2 a-4 b^3+b \left (8 a^2-39 b a-12 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\int \frac {8 a^4+40 b a^3+80 b^2 a^2+17 b^3 a+4 b^4+b \left (8 a^3+32 b a^2-15 b^2 a-4 b^3\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a+b\right )}d\tan (e+f x)}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\frac {8 (a+b)^4 \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)}{a}-\frac {b^3 \left (63 a^2+36 a b+8 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {-\frac {-\frac {\frac {8 (a+b)^4 \arctan (\tan (e+f x))}{a}-\frac {b^3 \left (63 a^2+36 a b+8 b^2\right ) \int \frac {1}{b \tan ^2(e+f x)+a+b}d\tan (e+f x)}{a}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {-\frac {\left (8 a^2-39 a b-12 b^2\right ) \cot ^3(e+f x)}{3 (a+b)}-\frac {-\frac {\frac {8 (a+b)^4 \arctan (\tan (e+f x))}{a}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \cot (e+f x)}{a+b}}{a+b}}{2 a (a+b)}-\frac {b (11 a+4 b) \cot ^3(e+f x)}{2 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )}}{4 a (a+b)}-\frac {b \cot ^3(e+f x)}{4 a (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}}{f}\)

Input:

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

Output:

(-1/4*(b*Cot[e + f*x]^3)/(a*(a + b)*(a + b + b*Tan[e + f*x]^2)^2) + ((-1/3 
*((8*a^2 - 39*a*b - 12*b^2)*Cot[e + f*x]^3)/(a + b) - (-(((8*(a + b)^4*Arc 
Tan[Tan[e + f*x]])/a - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTan[(Sqrt[b]* 
Tan[e + f*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b)) - ((8*a^3 + 32*a^2*b 
 - 15*a*b^2 - 4*b^3)*Cot[e + f*x])/(a + b))/(a + b))/(2*a*(a + b)) - (b*(1 
1*a + 4*b)*Cot[e + f*x]^3)/(2*a*(a + b)*(a + b + b*Tan[e + f*x]^2)))/(4*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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, 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 2075

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa 
ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi 
alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] &&  ! 
BinomialMatchQ[{u, v}, x]
 

rule 3042

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

rule 4629

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f 
_.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim 
p[ff/f   Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 
)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte 
gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
 

Maple [A] (verified)

Time = 22.67 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{3}}-\frac {1}{3 \left (a +b \right )^{3} \tan \left (f x +e \right )^{3}}-\frac {-a -4 b}{\left (a +b \right )^{4} \tan \left (f x +e \right )}-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{2} b +\frac {1}{2} a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (17 a^{2}+21 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{3} \left (a +b \right )^{4}}}{f}\)	\(171\)
default	\(\frac {\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{3}}-\frac {1}{3 \left (a +b \right )^{3} \tan \left (f x +e \right )^{3}}-\frac {-a -4 b}{\left (a +b \right )^{4} \tan \left (f x +e \right )}-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{2} b +\frac {1}{2} a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (17 a^{2}+21 a b +4 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{3} \left (a +b \right )^{4}}}{f}\)	\(171\)
risch	\(\text {Expression too large to display}\)	\(1015\)

Input:

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

Output:

1/f*(1/a^3*arctan(tan(f*x+e))-1/3/(a+b)^3/tan(f*x+e)^3-(-a-4*b)/(a+b)^4/ta 
n(f*x+e)-b^3/a^3/(a+b)^4*(((15/8*a^2*b+1/2*a*b^2)*tan(f*x+e)^3+1/8*a*(17*a 
^2+21*a*b+4*b^2)*tan(f*x+e))/(a+b+b*tan(f*x+e)^2)^2+1/8*(63*a^2+36*a*b+8*b 
^2)/((a+b)*b)^(1/2)*arctan(b*tan(f*x+e)/((a+b)*b)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (212) = 424\).

Time = 0.19 (sec) , antiderivative size = 1649, normalized size of antiderivative = 7.17 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/96*(4*(32*a^6 + 104*a^5*b + 51*a^3*b^3 + 18*a^2*b^4)*cos(f*x + e)^7 - 4 
*(24*a^6 + 32*a^5*b - 208*a^4*b^2 + 102*a^3*b^3 - 9*a^2*b^4 - 12*a*b^5)*co 
s(f*x + e)^5 - 4*(48*a^5*b + 160*a^4*b^2 - 155*a^3*b^3 + 72*a^2*b^4 + 24*a 
*b^5)*cos(f*x + e)^3 + 3*((63*a^4*b^2 + 36*a^3*b^3 + 8*a^2*b^4)*cos(f*x + 
e)^6 - 63*a^2*b^4 - 36*a*b^5 - 8*b^6 - (63*a^4*b^2 - 90*a^3*b^3 - 64*a^2*b 
^4 - 16*a*b^5)*cos(f*x + e)^4 - (126*a^3*b^3 + 9*a^2*b^4 - 20*a*b^5 - 8*b^ 
6)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e 
)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x 
+ e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a 
^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(8*a^4* 
b^2 + 32*a^3*b^3 - 15*a^2*b^4 - 4*a*b^5)*cos(f*x + e) + 96*((a^6 + 4*a^5*b 
 + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*x*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 
2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*x*cos(f*x + e)^4 - (2*a^5*b 
 + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*x*cos(f*x + e)^2 - 
 (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*x)*sin(f*x + e))/(((a 
^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 
2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 
- (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*co 
s(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)* 
sin(f*x + e)), 1/48*(2*(32*a^6 + 104*a^5*b + 51*a^3*b^3 + 18*a^2*b^4)*c...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.78 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 36 \, a b^{4} + 8 \, b^{5}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {{\left (a + b\right )} b}} - \frac {3 \, {\left (8 \, a^{3} b^{2} + 32 \, a^{2} b^{3} - 15 \, a b^{4} - 4 \, b^{5}\right )} \tan \left (f x + e\right )^{6} - 8 \, a^{5} - 24 \, a^{4} b - 24 \, a^{3} b^{2} - 8 \, a^{2} b^{3} + {\left (48 \, a^{4} b + 232 \, a^{3} b^{2} + 133 \, a^{2} b^{3} - 63 \, a b^{4} - 12 \, b^{5}\right )} \tan \left (f x + e\right )^{4} + 8 \, {\left (3 \, a^{5} + 16 \, a^{4} b + 23 \, a^{3} b^{2} + 10 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{6} b^{2} + 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} + 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \tan \left (f x + e\right )^{7} + 2 \, {\left (a^{7} b + 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} + a^{2} b^{6}\right )} \tan \left (f x + e\right )^{5} + {\left (a^{8} + 6 \, a^{7} b + 15 \, a^{6} b^{2} + 20 \, a^{5} b^{3} + 15 \, a^{4} b^{4} + 6 \, a^{3} b^{5} + a^{2} b^{6}\right )} \tan \left (f x + e\right )^{3}} - \frac {24 \, {\left (f x + e\right )}}{a^{3}}}{24 \, f} \] Input:

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

Output:

-1/24*(3*(63*a^2*b^3 + 36*a*b^4 + 8*b^5)*arctan(b*tan(f*x + e)/sqrt((a + b 
)*b))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt((a + b)*b)) 
- (3*(8*a^3*b^2 + 32*a^2*b^3 - 15*a*b^4 - 4*b^5)*tan(f*x + e)^6 - 8*a^5 - 
24*a^4*b - 24*a^3*b^2 - 8*a^2*b^3 + (48*a^4*b + 232*a^3*b^2 + 133*a^2*b^3 
- 63*a*b^4 - 12*b^5)*tan(f*x + e)^4 + 8*(3*a^5 + 16*a^4*b + 23*a^3*b^2 + 1 
0*a^2*b^3)*tan(f*x + e)^2)/((a^6*b^2 + 4*a^5*b^3 + 6*a^4*b^4 + 4*a^3*b^5 + 
 a^2*b^6)*tan(f*x + e)^7 + 2*(a^7*b + 5*a^6*b^2 + 10*a^5*b^3 + 10*a^4*b^4 
+ 5*a^3*b^5 + a^2*b^6)*tan(f*x + e)^5 + (a^8 + 6*a^7*b + 15*a^6*b^2 + 20*a 
^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6)*tan(f*x + e)^3) - 24*(f*x + e)/ 
a^3)/f
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 36 \, a b^{4} + 8 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {a b + b^{2}}} + \frac {3 \, {\left (15 \, a b^{4} \tan \left (f x + e\right )^{3} + 4 \, b^{5} \tan \left (f x + e\right )^{3} + 17 \, a^{2} b^{3} \tan \left (f x + e\right ) + 21 \, a b^{4} \tan \left (f x + e\right ) + 4 \, b^{5} \tan \left (f x + e\right )\right )}}{{\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac {24 \, {\left (f x + e\right )}}{a^{3}} - \frac {8 \, {\left (3 \, a \tan \left (f x + e\right )^{2} + 12 \, b \tan \left (f x + e\right )^{2} - a - b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (f x + e\right )^{3}}}{24 \, f} \] Input:

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

Output:

-1/24*(3*(63*a^2*b^3 + 36*a*b^4 + 8*b^5)*(pi*floor((f*x + e)/pi + 1/2)*sgn 
(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))/((a^7 + 4*a^6*b + 6*a^5*b^2 
+ 4*a^4*b^3 + a^3*b^4)*sqrt(a*b + b^2)) + 3*(15*a*b^4*tan(f*x + e)^3 + 4*b 
^5*tan(f*x + e)^3 + 17*a^2*b^3*tan(f*x + e) + 21*a*b^4*tan(f*x + e) + 4*b^ 
5*tan(f*x + e))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*(b*tan( 
f*x + e)^2 + a + b)^2) - 24*(f*x + e)/a^3 - 8*(3*a*tan(f*x + e)^2 + 12*b*t 
an(f*x + e)^2 - a - b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*tan(f* 
x + e)^3))/f
 

Mupad [B] (verification not implemented)

Time = 22.91 (sec) , antiderivative size = 7057, normalized size of antiderivative = 30.68 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((860160*a^6*b^20*tan(e + f*x))/(860160*a^6*b^20 + 14515200*a^7*b^19 + 
 115347456*a^8*b^18 + 570587136*a^9*b^17 + 1961717760*a^10*b^16 + 49658112 
00*a^11*b^15 + 9577308160*a^12*b^14 + 14379552768*a^13*b^13 + 17038737408* 
a^14*b^12 + 16066462720*a^15*b^11 + 12106321920*a^16*b^10 + 7294187520*a^1 
7*b^9 + 3502829568*a^18*b^8 + 1329527808*a^19*b^7 + 392232960*a^20*b^6 + 8 
7162880*a^21*b^5 + 13762560*a^22*b^4 + 1376256*a^23*b^3 + 65536*a^24*b^2) 
+ (14515200*a^7*b^19*tan(e + f*x))/(860160*a^6*b^20 + 14515200*a^7*b^19 + 
115347456*a^8*b^18 + 570587136*a^9*b^17 + 1961717760*a^10*b^16 + 496581120 
0*a^11*b^15 + 9577308160*a^12*b^14 + 14379552768*a^13*b^13 + 17038737408*a 
^14*b^12 + 16066462720*a^15*b^11 + 12106321920*a^16*b^10 + 7294187520*a^17 
*b^9 + 3502829568*a^18*b^8 + 1329527808*a^19*b^7 + 392232960*a^20*b^6 + 87 
162880*a^21*b^5 + 13762560*a^22*b^4 + 1376256*a^23*b^3 + 65536*a^24*b^2) + 
 (115347456*a^8*b^18*tan(e + f*x))/(860160*a^6*b^20 + 14515200*a^7*b^19 + 
115347456*a^8*b^18 + 570587136*a^9*b^17 + 1961717760*a^10*b^16 + 496581120 
0*a^11*b^15 + 9577308160*a^12*b^14 + 14379552768*a^13*b^13 + 17038737408*a 
^14*b^12 + 16066462720*a^15*b^11 + 12106321920*a^16*b^10 + 7294187520*a^17 
*b^9 + 3502829568*a^18*b^8 + 1329527808*a^19*b^7 + 392232960*a^20*b^6 + 87 
162880*a^21*b^5 + 13762560*a^22*b^4 + 1376256*a^23*b^3 + 65536*a^24*b^2) + 
 (570587136*a^9*b^17*tan(e + f*x))/(860160*a^6*b^20 + 14515200*a^7*b^19 + 
115347456*a^8*b^18 + 570587136*a^9*b^17 + 1961717760*a^10*b^16 + 496581...
 

Reduce [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 2154, normalized size of antiderivative = 9.37 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 189*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/ 
sqrt(b))*sin(e + f*x)**7*a**4*b**2 - 108*sqrt(b)*sqrt(a + b)*atan((sqrt(a 
+ b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**7*a**3*b**3 - 24*s 
qrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))* 
sin(e + f*x)**7*a**2*b**4 + 378*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan( 
(e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**5*a**4*b**2 + 594*sqrt(b)*s 
qrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + 
f*x)**5*a**3*b**3 + 264*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x 
)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**5*a**2*b**4 + 48*sqrt(b)*sqrt(a + b 
)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**5*a 
*b**5 - 189*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt( 
a))/sqrt(b))*sin(e + f*x)**3*a**4*b**2 - 486*sqrt(b)*sqrt(a + b)*atan((sqr 
t(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**3*a**3*b**3 - 
429*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt 
(b))*sin(e + f*x)**3*a**2*b**4 - 156*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b) 
*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + f*x)**3*a*b**5 - 24*sqrt(b)* 
sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) - sqrt(a))/sqrt(b))*sin(e + 
 f*x)**3*b**6 - 189*sqrt(b)*sqrt(a + b)*atan((sqrt(a + b)*tan((e + f*x)/2) 
 + sqrt(a))/sqrt(b))*sin(e + f*x)**7*a**4*b**2 - 108*sqrt(b)*sqrt(a + b)*a 
tan((sqrt(a + b)*tan((e + f*x)/2) + sqrt(a))/sqrt(b))*sin(e + f*x)**7*a...