Integrand size = 25, antiderivative size = 327 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^5 (a-b)^2 f}+\frac {\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^4 (a-b)^2 f}-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 f} \]
-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 *(11*a-8*b)*b*cot(f*x+e)^5/a^2/(a-b)^2/f/(a+b*tan(f*x+e)^2)^(1/2)-1/15*(15 *a^4+10*a^3*b+8*a^2*b^2-176*a*b^3+128*b^4)*cot(f*x+e)*(a+b*tan(f*x+e)^2)^( 1/2)/a^5/(a-b)^2/f+1/15*(5*a^3+4*a^2*b-88*a*b^2+64*b^3)*cot(f*x+e)^3*(a+b* tan(f*x+e)^2)^(1/2)/a^4/(a-b)^2/f-1/5*(a^2-22*a*b+16*b^2)*cot(f*x+e)^5*(a+ b*tan(f*x+e)^2)^(1/2)/a^3/(a-b)^2/f-1/3*b*cot(f*x+e)^5/a/(a-b)/f/(a+b*tan( f*x+e)^2)^(3/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 17.74 (sec) , antiderivative size = 921, normalized size of antiderivative = 2.82 \[ \int \frac {\cot ^6(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 {\left (-23 a^2 \cos (e+f x)-54 a b \cos (e+f x)-73 b^2 \cos (e+f x)\right ) \csc (e+f x)}{15 a^5}+\frac {(11 a \cos (e+f x)+14 b \cos (e+f x)) \csc ^3(e+f x)}{15 a^4}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a^3}-\frac {2 b^5 \sin (2 (e+f x))}{3 a^4 (a-b)^2 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}+\frac {15 a b^4 \sin (2 (e+f x))-11 b^5 \sin (2 (e+f x))}{3 a^5 (a-b)^2 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}\right )}{f} \]
-((-((b*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*Sq rt[-((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)*Co s[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)^2*f)) + (Sqrt[(a + b + a*Cos[ 2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-23*a^2*Cos[ e + f*x] - 54*a*b*Cos[e + f*x] - 73*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^ 5) + ((11*a*Cos[e + f*x] + 14*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^4) ...
Time = 0.60 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 4153, 374, 441, 27, 445, 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.
| \(\displaystyle \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^6 \left (a+b \tan (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^6(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{5/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\frac {\int \frac {\cot ^6(e+f x) \left (-8 b \tan ^2(e+f x)+3 a-8 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan (e+f x)}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 \cot ^6(e+f x) \left (a^2-22 b a+16 b^2-2 (11 a-8 b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\cot ^6(e+f x) \left (a^2-22 b a+16 b^2-2 (11 a-8 b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {\cot ^4(e+f x) \left (5 a^3+4 b a^2-88 b^2 a+64 b^3+4 b \left (a^2-22 b a+16 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{5 a}-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a}\right )}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {\int \frac {\cot ^2(e+f x) \left (15 a^4+10 b a^3+8 b^2 a^2-176 b^3 a+128 b^4+2 b \left (5 a^3+4 b a^2-88 b^2 a+64 b^3\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{3 a}-\frac {\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a}}{5 a}-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a}\right )}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {\int \frac {15 a^5}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{a}-\frac {\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{3 a}-\frac {\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a}}{5 a}-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a}\right )}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-15 a^4 \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)-\frac {\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{3 a}-\frac {\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a}}{5 a}-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a}\right )}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-15 a^4 \int \frac {1}{1-\frac {(b-a) \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}-\frac {\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{3 a}-\frac {\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a}}{5 a}-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a}\right )}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 a}-\frac {-\frac {\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a}-\frac {-\frac {15 a^4 \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{\sqrt {a-b}}-\frac {\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{3 a}}{5 a}\right )}{a (a-b)}-\frac {b (11 a-8 b) \cot ^5(e+f x)}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{3 a (a-b)}-\frac {b \cot ^5(e+f x)}{3 a (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}}{f}\) |
(-1/3*(b*Cot[e + f*x]^5)/(a*(a - b)*(a + b*Tan[e + f*x]^2)^(3/2)) + (-(((1 1*a - 8*b)*b*Cot[e + f*x]^5)/(a*(a - b)*Sqrt[a + b*Tan[e + f*x]^2])) + (3* (-1/5*((a^2 - 22*a*b + 16*b^2)*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/ a - (-1/3*((5*a^3 + 4*a^2*b - 88*a*b^2 + 64*b^3)*Cot[e + f*x]^3*Sqrt[a + b *Tan[e + f*x]^2])/a - ((-15*a^4*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/Sqrt[a - b] - ((15*a^4 + 10*a^3*b + 8*a^2*b^2 - 176*a *b^3 + 128*b^4)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/(3*a))/(5*a))) /(a*(a - b)))/(3*a*(a - b)))/f
3.4.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
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]))
\[\int \frac {\cot \left (f x +e \right )^{6}}{\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}d x\]
Time = 0.42 (sec) , antiderivative size = 1023, normalized size of antiderivative = 3.13 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/60*(15*(a^5*b^2*tan(f*x + e)^9 + 2*a^6*b*tan(f*x + e)^7 + a^7*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^5*b^2 - 5*a^4*b^3 - 2*a^3*b^4 - 184*a^2*b^5 + 304* a*b^6 - 128*b^7)*tan(f*x + e)^8 + 3*a^7 - 9*a^6*b + 9*a^5*b^2 - 3*a^4*b^3 + 3*(10*a^6*b - 5*a^5*b^2 - a^4*b^3 - 92*a^3*b^4 + 152*a^2*b^5 - 64*a*b^6) *tan(f*x + e)^6 + 3*(5*a^7 - 5*a^6*b + a^5*b^2 - 23*a^4*b^3 + 38*a^3*b^4 - 16*a^2*b^5)*tan(f*x + e)^4 - (5*a^7 - 7*a^6*b - 9*a^5*b^2 + 19*a^4*b^3 - 8*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b ^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^9 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7* b^3 - a^6*b^4)*f*tan(f*x + e)^7 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f *tan(f*x + e)^5), -1/30*(15*(a^5*b^2*tan(f*x + e)^9 + 2*a^6*b*tan(f*x + e) ^7 + a^7*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^5*b^2 - 5*a^4*b^3 - 2*a^3*b^4 - 184*a^2*b^5 + 304*a*b^6 - 128*b^7)*tan(f*x + e)^ 8 + 3*a^7 - 9*a^6*b + 9*a^5*b^2 - 3*a^4*b^3 + 3*(10*a^6*b - 5*a^5*b^2 - a^ 4*b^3 - 92*a^3*b^4 + 152*a^2*b^5 - 64*a*b^6)*tan(f*x + e)^6 + 3*(5*a^7 - 5 *a^6*b + a^5*b^2 - 23*a^4*b^3 + 38*a^3*b^4 - 16*a^2*b^5)*tan(f*x + e)^4 - (5*a^7 - 7*a^6*b - 9*a^5*b^2 + 19*a^4*b^3 - 8*a^3*b^4)*tan(f*x + e)^2)*...
\[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{6}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \]