Integrand size = 23, antiderivative size = 218 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {x}{(a-b)^2}+\frac {(9 a-7 b) b^{7/2} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{9/2} (a-b)^2 f}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{2 a^4 (a-b) f}+\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 (a-b) f}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 (a-b) f}-\frac {b \cot ^5(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )} \]
-x/(a-b)^2+1/2*(9*a-7*b)*b^(7/2)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))/a^(9/2 )/(a-b)^2/f-1/2*(2*a^3+2*a^2*b+2*a*b^2-7*b^3)*cot(f*x+e)/a^4/(a-b)/f+1/6*( 2*a^2+2*a*b-7*b^2)*cot(f*x+e)^3/a^3/(a-b)/f-1/10*(2*a-7*b)*cot(f*x+e)^5/a^ 2/(a-b)/f-1/2*b*cot(f*x+e)^5/a/(a-b)/f/(a+b*tan(f*x+e)^2)
Time = 5.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\frac {15 (9 a-7 b) b^{7/2} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{9/2} (a-b)^2}-\frac {2 \cot (e+f x) \left (23 a^2+40 a b+45 b^2-a (11 a+10 b) \csc ^2(e+f x)+3 a^2 \csc ^4(e+f x)\right )}{a^4}+\frac {15 \left (-2 (e+f x)+\frac {(a-b) b^4 \sin (2 (e+f x))}{a^4 (a+b+(a-b) \cos (2 (e+f x)))}\right )}{(a-b)^2}}{30 f} \]
((15*(9*a - 7*b)*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(9/2)* (a - b)^2) - (2*Cot[e + f*x]*(23*a^2 + 40*a*b + 45*b^2 - a*(11*a + 10*b)*C sc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4))/a^4 + (15*(-2*(e + f*x) + ((a - b)* b^4*Sin[2*(e + f*x)])/(a^4*(a + b + (a - b)*Cos[2*(e + f*x)]))))/(a - b)^2 )/(30*f)
Time = 0.48 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4153, 374, 445, 27, 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.
| \(\displaystyle \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^6 \left (a+b \tan (e+f x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot ^6(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\frac {\int \frac {\cot ^6(e+f x) \left (-7 b \tan ^2(e+f x)+2 a-7 b\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {5 \cot ^4(e+f x) \left (2 a^2+2 b a-7 b^2+(2 a-7 b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{5 a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\cot ^4(e+f x) \left (2 a^2+2 b a-7 b^2+(2 a-7 b) b \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {3 \cot ^2(e+f x) \left (2 a^3+2 b a^2+2 b^2 a-7 b^3+b \left (2 a^2+2 b a-7 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{3 a}-\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\int \frac {\cot ^2(e+f x) \left (2 a^3+2 b a^2+2 b^2 a-7 b^3+b \left (2 a^2+2 b a-7 b^2\right ) \tan ^2(e+f x)\right )}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{a}-\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {-\frac {\int \frac {2 a^4+2 b a^3+2 b^2 a^2+2 b^3 a-7 b^4+b \left (2 a^3+2 b a^2+2 b^2 a-7 b^3\right ) \tan ^2(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )}d\tan (e+f x)}{a}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{a}}{a}-\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {-\frac {\frac {2 a^4 \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)}{a-b}-\frac {b^4 (9 a-7 b) \int \frac {1}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{a-b}}{a}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{a}}{a}-\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {-\frac {\frac {2 a^4 \arctan (\tan (e+f x))}{a-b}-\frac {b^4 (9 a-7 b) \int \frac {1}{b \tan ^2(e+f x)+a}d\tan (e+f x)}{a-b}}{a}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{a}}{a}-\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{3 a}}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{3 a}-\frac {-\frac {\frac {2 a^4 \arctan (\tan (e+f x))}{a-b}-\frac {b^{7/2} (9 a-7 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)}}{a}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{a}}{a}}{a}-\frac {(2 a-7 b) \cot ^5(e+f x)}{5 a}}{2 a (a-b)}-\frac {b \cot ^5(e+f x)}{2 a (a-b) \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
((-1/5*((2*a - 7*b)*Cot[e + f*x]^5)/a - (-1/3*((2*a^2 + 2*a*b - 7*b^2)*Cot [e + f*x]^3)/a - (-(((2*a^4*ArcTan[Tan[e + f*x]])/(a - b) - ((9*a - 7*b)*b ^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*(a - b)))/a) - ((2 *a^3 + 2*a^2*b + 2*a*b^2 - 7*b^3)*Cot[e + f*x])/a)/a)/a)/(2*a*(a - b)) - ( b*Cot[e + f*x]^5)/(2*a*(a - b)*(a + b*Tan[e + f*x]^2)))/f
3.3.36.3.1 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[((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]
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[((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]
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]))
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{5 a^{2} \tan \left (f x +e \right )^{5}}-\frac {-2 b -a}{3 a^{3} \tan \left (f x +e \right )^{3}}-\frac {a^{2}+2 a b +3 b^{2}}{a^{4} \tan \left (f x +e \right )}+\frac {b^{4} \left (\frac {\left (\frac {a}{2}-\frac {b}{2}\right ) \tan \left (f x +e \right )}{a +b \tan \left (f x +e \right )^{2}}+\frac {\left (9 a -7 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4} \left (a -b \right )^{2}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{2}}}{f}\) | \(152\) |
| default | \(\frac {-\frac {1}{5 a^{2} \tan \left (f x +e \right )^{5}}-\frac {-2 b -a}{3 a^{3} \tan \left (f x +e \right )^{3}}-\frac {a^{2}+2 a b +3 b^{2}}{a^{4} \tan \left (f x +e \right )}+\frac {b^{4} \left (\frac {\left (\frac {a}{2}-\frac {b}{2}\right ) \tan \left (f x +e \right )}{a +b \tan \left (f x +e \right )^{2}}+\frac {\left (9 a -7 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4} \left (a -b \right )^{2}}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{2}}}{f}\) | \(152\) |
| risch | \(-\frac {x}{a^{2}-2 a b +b^{2}}-\frac {i \left (-60 a^{3} b^{2} {\mathrm e}^{10 i \left (f x +e \right )}-60 a^{2} b^{3} {\mathrm e}^{10 i \left (f x +e \right )}+205 a \,b^{4}-58 a^{4} b -76 a^{2} b^{3}-12 a^{3} b^{2}-105 b^{5}+46 a^{5}+208 a^{4} b \,{\mathrm e}^{2 i \left (f x +e \right )}-12 a^{3} b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+132 a^{2} b^{3} {\mathrm e}^{2 i \left (f x +e \right )}-880 a \,b^{4} {\mathrm e}^{2 i \left (f x +e \right )}-1720 a \,b^{4} {\mathrm e}^{6 i \left (f x +e \right )}-738 a^{4} b \,{\mathrm e}^{4 i \left (f x +e \right )}+168 a^{3} b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+344 a^{2} b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+1605 a \,b^{4} {\mathrm e}^{4 i \left (f x +e \right )}+420 a^{3} b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+500 a^{2} b^{3} {\mathrm e}^{8 i \left (f x +e \right )}+1255 a \,b^{4} {\mathrm e}^{8 i \left (f x +e \right )}+640 a^{4} b \,{\mathrm e}^{6 i \left (f x +e \right )}-120 a^{3} b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-840 a^{2} b^{3} {\mathrm e}^{6 i \left (f x +e \right )}-600 a \,b^{4} {\mathrm e}^{10 i \left (f x +e \right )}-910 a^{4} b \,{\mathrm e}^{8 i \left (f x +e \right )}-150 a^{4} b \,{\mathrm e}^{12 i \left (f x +e \right )}+135 a \,b^{4} {\mathrm e}^{12 i \left (f x +e \right )}+240 a^{4} b \,{\mathrm e}^{10 i \left (f x +e \right )}-1575 b^{5} {\mathrm e}^{8 i \left (f x +e \right )}-48 a^{5} {\mathrm e}^{2 i \left (f x +e \right )}-1575 b^{5} {\mathrm e}^{4 i \left (f x +e \right )}+46 a^{5} {\mathrm e}^{4 i \left (f x +e \right )}+2100 b^{5} {\mathrm e}^{6 i \left (f x +e \right )}+240 a^{5} {\mathrm e}^{6 i \left (f x +e \right )}+630 b^{5} {\mathrm e}^{2 i \left (f x +e \right )}+90 a^{5} {\mathrm e}^{12 i \left (f x +e \right )}-105 b^{5} {\mathrm e}^{12 i \left (f x +e \right )}+630 b^{5} {\mathrm e}^{10 i \left (f x +e \right )}+10 a^{5} {\mathrm e}^{8 i \left (f x +e \right )}\right )}{15 f \,a^{4} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5} \left (a -b \right )^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )}-\frac {9 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{4 a^{4} \left (a -b \right )^{2} f}+\frac {7 \sqrt {-a b}\, b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{4 a^{5} \left (a -b \right )^{2} f}+\frac {9 \sqrt {-a b}\, b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{4 a^{4} \left (a -b \right )^{2} f}-\frac {7 \sqrt {-a b}\, b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{4 a^{5} \left (a -b \right )^{2} f}\) | \(874\) |
1/f*(-1/5/a^2/tan(f*x+e)^5-1/3*(-2*b-a)/a^3/tan(f*x+e)^3-(a^2+2*a*b+3*b^2) /a^4/tan(f*x+e)+b^4/a^4/(a-b)^2*((1/2*a-1/2*b)*tan(f*x+e)/(a+b*tan(f*x+e)^ 2)+1/2*(9*a-7*b)/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2)))-1/(a-b)^2*a rctan(tan(f*x+e)))
Time = 0.36 (sec) , antiderivative size = 672, normalized size of antiderivative = 3.08 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\left [-\frac {120 \, a^{4} b f x \tan \left (f x + e\right )^{7} + 120 \, a^{5} f x \tan \left (f x + e\right )^{5} + 60 \, {\left (2 \, a^{4} b - 9 \, a b^{4} + 7 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + 24 \, a^{5} - 48 \, a^{4} b + 24 \, a^{3} b^{2} + 40 \, {\left (3 \, a^{5} - a^{4} b - 9 \, a^{2} b^{3} + 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - 8 \, {\left (5 \, a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2} + 15 \, {\left ({\left (9 \, a b^{4} - 7 \, b^{5}\right )} \tan \left (f x + e\right )^{7} + {\left (9 \, a^{2} b^{3} - 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{4} - 6 \, a b \tan \left (f x + e\right )^{2} + a^{2} - 4 \, {\left (a b \tan \left (f x + e\right )^{3} - a^{2} \tan \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right )}{120 \, {\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 {60 \, a^{4} b f x \tan \left (f x + e\right )^{7} + 60 \, a^{5} f x \tan \left (f x + e\right )^{5} + 30 \, {\left (2 \, a^{4} b - 9 \, a b^{4} + 7 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + 12 \, a^{5} - 24 \, a^{4} b + 12 \, a^{3} b^{2} + 20 \, {\left (3 \, a^{5} - a^{4} b - 9 \, a^{2} b^{3} + 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - 4 \, {\left (5 \, a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2} - 15 \, {\left ({\left (9 \, a b^{4} - 7 \, b^{5}\right )} \tan \left (f x + e\right )^{7} + {\left (9 \, a^{2} b^{3} - 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt {\frac {b}{a}}}{2 \, b \tan \left (f x + e\right )}\right )}{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 )}}\right ] \]
[-1/120*(120*a^4*b*f*x*tan(f*x + e)^7 + 120*a^5*f*x*tan(f*x + e)^5 + 60*(2 *a^4*b - 9*a*b^4 + 7*b^5)*tan(f*x + e)^6 + 24*a^5 - 48*a^4*b + 24*a^3*b^2 + 40*(3*a^5 - a^4*b - 9*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^4 - 8*(5*a^5 - 3*a ^4*b - 9*a^3*b^2 + 7*a^2*b^3)*tan(f*x + e)^2 + 15*((9*a*b^4 - 7*b^5)*tan(f *x + e)^7 + (9*a^2*b^3 - 7*a*b^4)*tan(f*x + e)^5)*sqrt(-b/a)*log((b^2*tan( f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan( f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/ ((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), -1/60*(60*a^4*b*f*x*tan(f*x + e)^7 + 60*a^5*f*x*tan(f *x + e)^5 + 30*(2*a^4*b - 9*a*b^4 + 7*b^5)*tan(f*x + e)^6 + 12*a^5 - 24*a^ 4*b + 12*a^3*b^2 + 20*(3*a^5 - a^4*b - 9*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^4 - 4*(5*a^5 - 3*a^4*b - 9*a^3*b^2 + 7*a^2*b^3)*tan(f*x + e)^2 - 15*((9*a*b ^4 - 7*b^5)*tan(f*x + e)^7 + (9*a^2*b^3 - 7*a*b^4)*tan(f*x + e)^5)*sqrt(b/ a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((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)]
Timed out. \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.10 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\frac {15 \, {\left (9 \, a b^{4} - 7 \, b^{5}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {a b}} - \frac {15 \, {\left (2 \, a^{3} b + 2 \, a^{2} b^{2} + 2 \, a b^{3} - 7 \, b^{4}\right )} \tan \left (f x + e\right )^{6} + 10 \, {\left (3 \, a^{4} + 2 \, a^{3} b + 2 \, a^{2} b^{2} - 7 \, a b^{3}\right )} \tan \left (f x + e\right )^{4} + 6 \, a^{4} - 6 \, a^{3} b - 2 \, {\left (5 \, a^{4} + 2 \, a^{3} b - 7 \, a^{2} b^{2}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{5} b - a^{4} b^{2}\right )} \tan \left (f x + e\right )^{7} + {\left (a^{6} - a^{5} b\right )} \tan \left (f x + e\right )^{5}} - \frac {30 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}}}{30 \, f} \]
1/30*(15*(9*a*b^4 - 7*b^5)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a^6 - 2*a^5* b + a^4*b^2)*sqrt(a*b)) - (15*(2*a^3*b + 2*a^2*b^2 + 2*a*b^3 - 7*b^4)*tan( f*x + e)^6 + 10*(3*a^4 + 2*a^3*b + 2*a^2*b^2 - 7*a*b^3)*tan(f*x + e)^4 + 6 *a^4 - 6*a^3*b - 2*(5*a^4 + 2*a^3*b - 7*a^2*b^2)*tan(f*x + e)^2)/((a^5*b - a^4*b^2)*tan(f*x + e)^7 + (a^6 - a^5*b)*tan(f*x + e)^5) - 30*(f*x + e)/(a ^2 - 2*a*b + b^2))/f
Time = 0.99 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {\frac {15 \, b^{4} \tan \left (f x + e\right )}{{\left (a^{5} - a^{4} b\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}} + \frac {15 \, {\left (9 \, a b^{4} - 7 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {a b}} - \frac {30 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {2 \, {\left (15 \, a^{2} \tan \left (f x + e\right )^{4} + 30 \, a b \tan \left (f x + e\right )^{4} + 45 \, b^{2} \tan \left (f x + e\right )^{4} - 5 \, a^{2} \tan \left (f x + e\right )^{2} - 10 \, a b \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )}}{a^{4} \tan \left (f x + e\right )^{5}}}{30 \, f} \]
1/30*(15*b^4*tan(f*x + e)/((a^5 - a^4*b)*(b*tan(f*x + e)^2 + a)) + 15*(9*a *b^4 - 7*b^5)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e) /sqrt(a*b)))/((a^6 - 2*a^5*b + a^4*b^2)*sqrt(a*b)) - 30*(f*x + e)/(a^2 - 2 *a*b + b^2) - 2*(15*a^2*tan(f*x + e)^4 + 30*a*b*tan(f*x + e)^4 + 45*b^2*ta n(f*x + e)^4 - 5*a^2*tan(f*x + e)^2 - 10*a*b*tan(f*x + e)^2 + 3*a^2)/(a^4* tan(f*x + e)^5))/f
Time = 14.94 (sec) , antiderivative size = 3030, normalized size of antiderivative = 13.90 \[ \int \frac {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
- (1/(5*a) + (tan(e + f*x)^4*(5*a*b + 3*a^2 + 7*b^2))/(3*a^3) - (tan(e + f *x)^2*(5*a + 7*b))/(15*a^2) + (tan(e + f*x)^6*(2*a*b^3 + 2*a^3*b - 7*b^4 + 2*a^2*b^2))/(2*a^4*(a - b)))/(f*(a*tan(e + f*x)^5 + b*tan(e + f*x)^7)) - (2*atan(((tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21* b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((2816*a^17*b^11 - 448*a^16*b^12 - 736 0*a^18*b^10 + 10240*a^19*b^9 - 8000*a^20*b^8 + 3200*a^21*b^7 + 64*a^22*b^6 - 1280*a^23*b^5 + 1280*a^24*b^4 - 640*a^25*b^3 + 128*a^26*b^2 + (tan(e + f*x)*(256*a^20*b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 3584 *a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2)*1i)/(2*a^2 - 4*a *b + 2*b^2))*1i)/(2*a^2 - 4*a*b + 2*b^2))/(2*a^2 - 4*a*b + 2*b^2) + (tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22* b^4 - 64*a^23*b^3) + ((448*a^16*b^12 - 2816*a^17*b^11 + 7360*a^18*b^10 - 1 0240*a^19*b^9 + 8000*a^20*b^8 - 3200*a^21*b^7 - 64*a^22*b^6 + 1280*a^23*b^ 5 - 1280*a^24*b^4 + 640*a^25*b^3 - 128*a^26*b^2 + (tan(e + f*x)*(256*a^20* b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 358 4*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2)*1i)/(2*a^2 - 4*a*b + 2*b^2))*1i )/(2*a^2 - 4*a*b + 2*b^2))/(2*a^2 - 4*a*b + 2*b^2))/(((tan(e + f*x)*(784*a ^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^1...