\(\int \frac {\cot ^6(e+f x)}{(a+b \tan ^2(e+f x))^{3/2}} \, dx\) [345]

Optimal result

Integrand size = 25, antiderivative size = 252 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{3/2} f}-\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (15 a^3+10 a^2 b+8 a b^2-48 b^3\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 (a-b) f}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 (a-b) f}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f} \]

[Out]

-arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))/(a-b)^(3/2)/f-b*cot(f*x+e)^5/a/(a-b)/f/(a+b*tan(f*x+e
)^2)^(1/2)-1/15*(15*a^3+10*a^2*b+8*a*b^2-48*b^3)*cot(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)/a^4/(a-b)/f+1/15*(5*a^2+4
*a*b-24*b^2)*cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2)/a^3/(a-b)/f-1/5*(a-6*b)*cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/
2)/a^2/(a-b)/f

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3751, 483, 597, 12, 385, 209} \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 f (a-b)}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 f (a-b)}-\frac {\left (15 a^3+10 a^2 b+8 a b^2-48 b^3\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)}-\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{3/2}}-\frac {b \cot ^5(e+f x)}{a f (a-b) \sqrt {a+b \tan ^2(e+f x)}} \]

[In]

Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - (b*Cot[e + f*x]^5)/(a*(a
- b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((15*a^3 + 10*a^2*b + 8*a*b^2 - 48*b^3)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f
*x]^2])/(15*a^4*(a - b)*f) + ((5*a^2 + 4*a*b - 24*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*(a -
b)*f) - ((a - 6*b)*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a^2*(a - b)*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

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

Rule 385

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

Rule 483

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

Rule 597

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

Rule 3751

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]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^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] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^6 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {a-6 b-6 b x^2}{x^6 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{a (a-b) f} \\ & = -\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f}-\frac {\text {Subst}\left (\int \frac {5 a^2+4 a b-24 b^2+4 (a-6 b) b x^2}{x^4 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{5 a^2 (a-b) f} \\ & = -\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 (a-b) f}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f}+\frac {\text {Subst}\left (\int \frac {15 a^3+10 a^2 b+8 a b^2-48 b^3+2 b \left (5 a^2+4 a b-24 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a^3 (a-b) f} \\ & = -\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (15 a^3+10 a^2 b+8 a b^2-48 b^3\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 (a-b) f}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 (a-b) f}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f}-\frac {\text {Subst}\left (\int \frac {15 a^4}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a^4 (a-b) f} \\ & = -\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (15 a^3+10 a^2 b+8 a b^2-48 b^3\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 (a-b) f}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 (a-b) f}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{(a-b) f} \\ & = -\frac {b \cot ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (15 a^3+10 a^2 b+8 a b^2-48 b^3\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 (a-b) f}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 (a-b) f}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f}-\frac {\text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b) f} \\ & = -\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 ^5(e+f x)}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (15 a^3+10 a^2 b+8 a b^2-48 b^3\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 (a-b) f}+\frac {\left (5 a^2+4 a b-24 b^2\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 (a-b) f}-\frac {(a-6 b) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^2 (a-b) f} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 17.56 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.37 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {-\frac {b \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{a (a+b+(a-b) \cos (2 (e+f x)))}-\frac {4 b \sqrt {1+\cos (2 (e+f x))} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{4 a \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}-\frac {\sqrt {-\frac {a \cot ^2(e+f x)}{b}} \sqrt {-\frac {a (1+\cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}} \csc (2 (e+f x)) \operatorname {EllipticPi}\left (-\frac {b}{a-b},\arcsin \left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right ),1\right ) \sin ^4(e+f x)}{2 (a-b) \sqrt {1+\cos (2 (e+f x))} \sqrt {a+b+(a-b) \cos (2 (e+f x))}}\right )}{\sqrt {a+b+(a-b) \cos (2 (e+f x))}}}{(a-b) f}+\frac {\sqrt {\frac {a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {\left (-23 a^2 \cos (e+f x)-34 a b \cos (e+f x)-33 b^2 \cos (e+f x)\right ) \csc (e+f x)}{15 a^4}+\frac {(11 a \cos (e+f x)+9 b \cos (e+f x)) \csc ^3(e+f x)}{15 a^3}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a^2}+\frac {b^4 \sin (2 (e+f x))}{a^4 (a-b) (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}\right )}{f} \]

[In]

Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]

[Out]

-((-((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]*Csc[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)*Sqrt[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)*f)) + (Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-2
3*a^2*Cos[e + f*x] - 34*a*b*Cos[e + f*x] - 33*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^4) + ((11*a*Cos[e + f*x] +
9*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a^2) + (b^4*Sin[2*(e + f*x)])/(
a^4*(a - b)*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))))/f

Maple [C] (warning: unable to verify)

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

Time = 5.33 (sec) , antiderivative size = 1971, normalized size of antiderivative = 7.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1971\)

[In]

int(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/480/f/a^4/((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)/(a-b)*(-656*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a
^2*b^2*(-cos(f*x+e)+1)^6*csc(f*x+e)^6-1280*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a*b^3*(-cos(f*x+e)+1)^6*c
sc(f*x+e)^6-51*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^3*b*(-cos(f*x+e)+1)^4*csc(f*x+e)^4+32*((2*I*b^(1/2)
*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^2*b^2*(-cos(f*x+e)+1)^4*csc(f*x+e)^4+384*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1
/2)*a*b^3*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-14*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^3*b*(-cos(f*x+e)+1)^2*
csc(f*x+e)^2-24*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^2*b^2*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+3*((2*I*b^(1/
2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^3*b*(-cos(f*x+e)+1)^12*csc(f*x+e)^12-14*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(
1/2)*a^3*b*(-cos(f*x+e)+1)^10*csc(f*x+e)^10+960*a^4*(-(2*I*(-cos(f*x+e)+1)^2*b^(1/2)*(a-b)^(1/2)*csc(f*x+e)^2+
a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2-2*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2-a)/a)^(1/2)*((2*I*(-cos(f*x+e)+1)^2*b^(1/2
)*(a-b)^(1/2)*csc(f*x+e)^2-a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+2*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)/a)^(1/2)*Ell
ipticF(((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))*(-cos(f*x+e)+1)^5*csc(f*x+e)^5-24*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)
/a)^(1/2)*a^2*b^2*(-cos(f*x+e)+1)^10*csc(f*x+e)^10-51*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^3*b*(-cos(f*
x+e)+1)^8*csc(f*x+e)^8+32*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^2*b^2*(-cos(f*x+e)+1)^8*csc(f*x+e)^8+384
*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a*b^3*(-cos(f*x+e)+1)^8*csc(f*x+e)^8-836*((2*I*b^(1/2)*(a-b)^(1/2)+
a-2*b)/a)^(1/2)*a^3*b*(-cos(f*x+e)+1)^6*csc(f*x+e)^6-3*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^4*(-cos(f*x
+e)+1)^12*csc(f*x+e)^12+38*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^4*(-cos(f*x+e)+1)^10*csc(f*x+e)^10-365*
((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^4*(-cos(f*x+e)+1)^8*csc(f*x+e)^8-3*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b
)/a)^(1/2)*a^4-1920*(-(2*I*(-cos(f*x+e)+1)^2*b^(1/2)*(a-b)^(1/2)*csc(f*x+e)^2+a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2
-2*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2-a)/a)^(1/2)*((2*I*(-cos(f*x+e)+1)^2*b^(1/2)*(a-b)^(1/2)*csc(f*x+e)^2-a*(-c
os(f*x+e)+1)^2*csc(f*x+e)^2+2*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)/a)^(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)/((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2))*a^4*(-cos(f*x+e)+1)^5*csc(f*x+e)^5+3*((2*I*b^(1/2)*(a-b
)^(1/2)+a-2*b)/a)^(1/2)*a^3*b+660*a^4*(-cos(f*x+e)+1)^6*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*csc(f*x+e)^6
-365*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*a^4*(-cos(f*x+e)+1)^4*csc(f*x+e)^4+38*((2*I*b^(1/2)*(a-b)^(1/2)
+a-2*b)/a)^(1/2)*a^4*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+3072*((2*I*b^(1/2)*(a-b)^(1/2)+a-2*b)/a)^(1/2)*b^4*(-cos(f
*x+e)+1)^6*csc(f*x+e)^6)*(a*(-cos(f*x+e)+1)^4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)
+1)^2*csc(f*x+e)^2+a)/(-cos(f*x+e)+1)^5*sin(f*x+e)^5/((-cos(f*x+e)+1)^2*csc(f*x+e)^2-1)^3/((a*(-cos(f*x+e)+1)^
4*csc(f*x+e)^4-2*a*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+4*b*(-cos(f*x+e)+1)^2*csc(f*x+e)^2+a)/((-cos(f*x+e)+1)^2*csc
(f*x+e)^2-1)^2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.73 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\left [\frac {15 \, {\left (a^{4} b \tan \left (f x + e\right )^{7} + a^{5} \tan \left (f x + e\right )^{5}\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 (15 \, a^{4} b - 5 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 56 \, a b^{4} + 48 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + 3 \, a^{5} - 6 \, a^{4} b + 3 \, a^{3} b^{2} + {\left (15 \, a^{5} - 10 \, a^{4} b - a^{3} b^{2} - 28 \, a^{2} b^{3} + 24 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{5} - 4 \, a^{4} b - 7 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{60 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{7} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{5}\right )}}, -\frac {15 \, {\left (a^{4} b \tan \left (f x + e\right )^{7} + a^{5} \tan \left (f x + e\right )^{5}\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 (15 \, a^{4} b - 5 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 56 \, a b^{4} + 48 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + 3 \, a^{5} - 6 \, a^{4} b + 3 \, a^{3} b^{2} + {\left (15 \, a^{5} - 10 \, a^{4} b - a^{3} b^{2} - 28 \, a^{2} b^{3} + 24 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{5} - 4 \, a^{4} b - 7 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{30 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{7} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{5}\right )}}\right ] \]

[In]

integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/60*(15*(a^4*b*tan(f*x + e)^7 + a^5*tan(f*x + e)^5)*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*((15*a^4*b - 5*a^3*b^2 - 2*a^2*b^3 - 56*a*b^4
+ 48*b^5)*tan(f*x + e)^6 + 3*a^5 - 6*a^4*b + 3*a^3*b^2 + (15*a^5 - 10*a^4*b - a^3*b^2 - 28*a^2*b^3 + 24*a*b^4
)*tan(f*x + e)^4 - (5*a^5 - 4*a^4*b - 7*a^3*b^2 + 6*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6
*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5), -1/30*(15*(a^4*b*tan
(f*x + e)^7 + a^5*tan(f*x + e)^5)*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*((15*a^4*b - 5*a^3*b^2 - 2*a^2*b^3 - 56*a*b^4 + 48*b^5)*tan(f*x + e)^6 + 3*a
^5 - 6*a^4*b + 3*a^3*b^2 + (15*a^5 - 10*a^4*b - a^3*b^2 - 28*a^2*b^3 + 24*a*b^4)*tan(f*x + e)^4 - (5*a^5 - 4*a
^4*b - 7*a^3*b^2 + 6*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan
(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5)]

Sympy [F]

\[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{6}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(cot(f*x+e)**6/(a+b*tan(f*x+e)**2)**(3/2),x)

[Out]

Integral(cot(e + f*x)**6/(a + b*tan(e + f*x)**2)**(3/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Hanged} \]

[In]

int(cot(e + f*x)^6/(a + b*tan(e + f*x)^2)^(3/2),x)

[Out]

\text{Hanged}