Integrand size = 23, antiderivative size = 182 \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {(3 a-7 b) (a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{9/2} f}-\frac {\left (5 a^2-20 a b+14 b^2\right ) \cot (e+f x)}{5 a^4 f}-\frac {(10 a-7 b) \cot ^3(e+f x)}{15 a^3 f}-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )}-\frac {b \left (5 a^2-10 a b+7 b^2\right ) \tan (e+f x)}{10 a^4 f \left (a+b \tan ^2(e+f x)\right )} \]
-1/5*(5*a^2-20*a*b+14*b^2)*cot(f*x+e)/a^4/f-1/15*(10*a-7*b)*cot(f*x+e)^3/a ^3/f-1/2*(3*a-7*b)*(a-b)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))*b^(1/2)/a^(9/2 )/f-1/5*cot(f*x+e)^5/a/f/(a+b*tan(f*x+e)^2)-1/10*b*(5*a^2-10*a*b+7*b^2)*ta n(f*x+e)/a^4/f/(a+b*tan(f*x+e)^2)
Time = 2.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.83 \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\frac {-15 \sqrt {b} \left (3 a^2-10 a b+7 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )+\sqrt {a} \left (-2 \cot (e+f x) \left (8 a^2-50 a b+45 b^2+2 a (2 a-5 b) \csc ^2(e+f x)+3 a^2 \csc ^4(e+f x)\right )-\frac {15 (a-b)^2 b \sin (2 (e+f x))}{a+b+(a-b) \cos (2 (e+f x))}\right )}{30 a^{9/2} f} \]
(-15*Sqrt[b]*(3*a^2 - 10*a*b + 7*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a ]] + Sqrt[a]*(-2*Cot[e + f*x]*(8*a^2 - 50*a*b + 45*b^2 + 2*a*(2*a - 5*b)*C sc[e + f*x]^2 + 3*a^2*Csc[e + f*x]^4) - (15*(a - b)^2*b*Sin[2*(e + f*x)])/ (a + b + (a - b)*Cos[2*(e + f*x)])))/(30*a^(9/2)*f)
Time = 0.43 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4146, 365, 361, 1584, 2009}
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.
| \(\displaystyle \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^6 \left (a+b \tan (e+f x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\cot ^6(e+f x) \left (\tan ^2(e+f x)+1\right )^2}{\left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {\frac {\int \frac {\cot ^4(e+f x) \left (5 a \tan ^2(e+f x)+10 a-7 b\right )}{\left (b \tan ^2(e+f x)+a\right )^2}d\tan (e+f x)}{5 a}-\frac {\cot ^5(e+f x)}{5 a \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} b \int \frac {\cot ^4(e+f x) \left (\frac {\left (5 a^2-10 b a+7 b^2\right ) \tan ^4(e+f x)}{a^3}+2 \left (-\frac {7 b}{a^2}+\frac {10}{a}-\frac {5}{b}\right ) \tan ^2(e+f x)+2 \left (\frac {7}{a}-\frac {10}{b}\right )\right )}{b \tan ^2(e+f x)+a}d\tan (e+f x)-\frac {b \left (5 a^2-10 a b+7 b^2\right ) \tan (e+f x)}{2 a^3 \left (a+b \tan ^2(e+f x)\right )}}{5 a}-\frac {\cot ^5(e+f x)}{5 a \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} b \int \left (-\frac {2 (10 a-7 b) \cot ^4(e+f x)}{a^2 b}-\frac {2 \left (5 a^2-20 b a+14 b^2\right ) \cot ^2(e+f x)}{a^3 b}+\frac {5 (3 a-7 b) (a-b)}{a^3 \left (b \tan ^2(e+f x)+a\right )}\right )d\tan (e+f x)-\frac {b \left (5 a^2-10 a b+7 b^2\right ) \tan (e+f x)}{2 a^3 \left (a+b \tan ^2(e+f x)\right )}}{5 a}-\frac {\cot ^5(e+f x)}{5 a \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {b \left (5 a^2-10 a b+7 b^2\right ) \tan (e+f x)}{2 a^3 \left (a+b \tan ^2(e+f x)\right )}-\frac {1}{2} b \left (\frac {5 (3 a-7 b) (a-b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}}+\frac {2 (10 a-7 b) \cot ^3(e+f x)}{3 a^2 b}+\frac {2 \left (5 a^2-20 a b+14 b^2\right ) \cot (e+f x)}{a^3 b}\right )}{5 a}-\frac {\cot ^5(e+f x)}{5 a \left (a+b \tan ^2(e+f x)\right )}}{f}\) |
(-1/5*Cot[e + f*x]^5/(a*(a + b*Tan[e + f*x]^2)) + (-1/2*(b*((5*(3*a - 7*b) *(a - b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*Sqrt[b]) + (2*(5 *a^2 - 20*a*b + 14*b^2)*Cot[e + f*x])/(a^3*b) + (2*(10*a - 7*b)*Cot[e + f* x]^3)/(3*a^2*b))) - (b*(5*a^2 - 10*a*b + 7*b^2)*Tan[e + f*x])/(2*a^3*(a + b*Tan[e + f*x]^2)))/(5*a))/f
3.1.79.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 1.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{5 a^{2} \tan \left (f x +e \right )^{5}}-\frac {2 a -2 b}{3 a^{3} \tan \left (f x +e \right )^{3}}-\frac {a^{2}-4 a b +3 b^{2}}{a^{4} \tan \left (f x +e \right )}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2}-a b +\frac {1}{2} b^{2}\right ) \tan \left (f x +e \right )}{a +b \tan \left (f x +e \right )^{2}}+\frac {\left (3 a^{2}-10 a b +7 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(144\) |
| default | \(\frac {-\frac {1}{5 a^{2} \tan \left (f x +e \right )^{5}}-\frac {2 a -2 b}{3 a^{3} \tan \left (f x +e \right )^{3}}-\frac {a^{2}-4 a b +3 b^{2}}{a^{4} \tan \left (f x +e \right )}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2}-a b +\frac {1}{2} b^{2}\right ) \tan \left (f x +e \right )}{a +b \tan \left (f x +e \right )^{2}}+\frac {\left (3 a^{2}-10 a b +7 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}}{f}\) | \(144\) |
| risch | \(\frac {i \left (2020 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+45 a^{2} b \,{\mathrm e}^{12 i \left (f x +e \right )}+65 a^{2} b \,{\mathrm e}^{8 i \left (f x +e \right )}-1480 a \,b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+690 a \,b^{2} {\mathrm e}^{10 i \left (f x +e \right )}-180 a^{2} b \,{\mathrm e}^{10 i \left (f x +e \right )}-150 a \,b^{2} {\mathrm e}^{12 i \left (f x +e \right )}+271 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-412 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}-1830 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+970 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+80 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}-220 a \,b^{2}+131 a^{2} b -16 a^{3}+105 b^{3}+1575 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+48 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-630 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}-2100 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}-16 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-160 a^{3} {\mathrm e}^{8 i \left (f x +e \right )}+1575 b^{3} {\mathrm e}^{8 i \left (f x +e \right )}-240 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+105 b^{3} {\mathrm e}^{12 i \left (f x +e \right )}-630 b^{3} {\mathrm e}^{10 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 \,{\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 {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{4 a^{3} f}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b}{2 a^{4} f}-\frac {7 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) b^{2}}{4 a^{5} f}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{4 a^{3} f}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b}{2 a^{4} f}+\frac {7 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) b^{2}}{4 a^{5} f}\) | \(714\) |
1/f*(-1/5/a^2/tan(f*x+e)^5-1/3*(2*a-2*b)/a^3/tan(f*x+e)^3-(a^2-4*a*b+3*b^2 )/a^4/tan(f*x+e)-1/a^4*b*((1/2*a^2-a*b+1/2*b^2)*tan(f*x+e)/(a+b*tan(f*x+e) ^2)+1/2*(3*a^2-10*a*b+7*b^2)/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^(1/2))) )
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (164) = 328\).
Time = 0.34 (sec) , antiderivative size = 855, normalized size of antiderivative = 4.70 \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\left [-\frac {4 \, {\left (16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right )} \cos \left (f x + e\right )^{7} - 4 \, {\left (40 \, a^{3} - 321 \, a^{2} b + 590 \, a b^{2} - 315 \, b^{3}\right )} \cos \left (f x + e\right )^{5} + 20 \, {\left (6 \, a^{3} - 47 \, a^{2} b + 104 \, a b^{2} - 63 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left ({\left (3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (6 \, a^{3} - 29 \, a^{2} b + 44 \, a b^{2} - 21 \, b^{3}\right )} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3} + {\left (3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) + 60 \, {\left (3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right )} \cos \left (f x + e\right )}{120 \, {\left ({\left (a^{5} - a^{4} b\right )} f \cos \left (f x + e\right )^{6} + a^{4} b f - {\left (2 \, a^{5} - 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{4} + {\left (a^{5} - 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, -\frac {2 \, {\left (16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (40 \, a^{3} - 321 \, a^{2} b + 590 \, a b^{2} - 315 \, b^{3}\right )} \cos \left (f x + e\right )^{5} + 10 \, {\left (6 \, a^{3} - 47 \, a^{2} b + 104 \, a b^{2} - 63 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left ({\left (3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )^{6} - {\left (6 \, a^{3} - 29 \, a^{2} b + 44 \, a b^{2} - 21 \, b^{3}\right )} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3} + {\left (3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 30 \, {\left (3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right )} \cos \left (f x + e\right )}{60 \, {\left ({\left (a^{5} - a^{4} b\right )} f \cos \left (f x + e\right )^{6} + a^{4} b f - {\left (2 \, a^{5} - 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{4} + {\left (a^{5} - 3 \, a^{4} b\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ] \]
[-1/120*(4*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 4*( 40*a^3 - 321*a^2*b + 590*a*b^2 - 315*b^3)*cos(f*x + e)^5 + 20*(6*a^3 - 47* a^2*b + 104*a*b^2 - 63*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 13*a^2*b + 17*a* b^2 - 7*b^3)*cos(f*x + e)^6 - (6*a^3 - 29*a^2*b + 44*a*b^2 - 21*b^3)*cos(f *x + e)^4 + 3*a^2*b - 10*a*b^2 + 7*b^3 + (3*a^3 - 19*a^2*b + 37*a*b^2 - 21 *b^3)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos( f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e) ^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(3*a^2*b - 10* a*b^2 + 7*b^3)*cos(f*x + e))/(((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2)*sin (f*x + e)), -1/60*(2*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 2*(40*a^3 - 321*a^2*b + 590*a*b^2 - 315*b^3)*cos(f*x + e)^5 + 10*(6 *a^3 - 47*a^2*b + 104*a*b^2 - 63*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 13*a^2 *b + 17*a*b^2 - 7*b^3)*cos(f*x + e)^6 - (6*a^3 - 29*a^2*b + 44*a*b^2 - 21* b^3)*cos(f*x + e)^4 + 3*a^2*b - 10*a*b^2 + 7*b^3 + (3*a^3 - 19*a^2*b + 37* a*b^2 - 21*b^3)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e) ^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(3*a^2* b - 10*a*b^2 + 7*b^3)*cos(f*x + e))/(((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4 *b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x +...
Timed out. \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {15 \, {\left (3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right )} \tan \left (f x + e\right )^{6} + 10 \, {\left (3 \, a^{3} - 10 \, a^{2} b + 7 \, a b^{2}\right )} \tan \left (f x + e\right )^{4} + 6 \, a^{3} + 2 \, {\left (10 \, a^{3} - 7 \, a^{2} b\right )} \tan \left (f x + e\right )^{2}}{a^{4} b \tan \left (f x + e\right )^{7} + a^{5} \tan \left (f x + e\right )^{5}} + \frac {15 \, {\left (3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}}}{30 \, f} \]
-1/30*((15*(3*a^2*b - 10*a*b^2 + 7*b^3)*tan(f*x + e)^6 + 10*(3*a^3 - 10*a^ 2*b + 7*a*b^2)*tan(f*x + e)^4 + 6*a^3 + 2*(10*a^3 - 7*a^2*b)*tan(f*x + e)^ 2)/(a^4*b*tan(f*x + e)^7 + a^5*tan(f*x + e)^5) + 15*(3*a^2*b - 10*a*b^2 + 7*b^3)*arctan(b*tan(f*x + e)/sqrt(a*b))/(sqrt(a*b)*a^4))/f
Time = 0.63 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {15 \, {\left (3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{\sqrt {a b} a^{4}} + \frac {15 \, {\left (a^{2} b \tan \left (f x + e\right ) - 2 \, a b^{2} \tan \left (f x + e\right ) + b^{3} \tan \left (f x + e\right )\right )}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )} a^{4}} + \frac {2 \, {\left (15 \, a^{2} \tan \left (f x + e\right )^{4} - 60 \, a b \tan \left (f x + e\right )^{4} + 45 \, b^{2} \tan \left (f x + e\right )^{4} + 10 \, a^{2} \tan \left (f x + e\right )^{2} - 10 \, a b \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )}}{a^{4} \tan \left (f x + e\right )^{5}}}{30 \, f} \]
-1/30*(15*(3*a^2*b - 10*a*b^2 + 7*b^3)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b ) + arctan(b*tan(f*x + e)/sqrt(a*b)))/(sqrt(a*b)*a^4) + 15*(a^2*b*tan(f*x + e) - 2*a*b^2*tan(f*x + e) + b^3*tan(f*x + e))/((b*tan(f*x + e)^2 + a)*a^ 4) + 2*(15*a^2*tan(f*x + e)^4 - 60*a*b*tan(f*x + e)^4 + 45*b^2*tan(f*x + e )^4 + 10*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 = 11.69 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98 \[ \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx=-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (3\,a^2-10\,a\,b+7\,b^2\right )}{3\,a^3}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (10\,a-7\,b\right )}{15\,a^2}+\frac {b\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (3\,a^2-10\,a\,b+7\,b^2\right )}{2\,a^4}}{f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^7+a\,{\mathrm {tan}\left (e+f\,x\right )}^5\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )\,\left (a-b\right )\,\left (3\,a-7\,b\right )}{\sqrt {a}\,\left (3\,a^2-10\,a\,b+7\,b^2\right )}\right )\,\left (a-b\right )\,\left (3\,a-7\,b\right )}{2\,a^{9/2}\,f} \]
- (1/(5*a) + (tan(e + f*x)^4*(3*a^2 - 10*a*b + 7*b^2))/(3*a^3) + (tan(e + f*x)^2*(10*a - 7*b))/(15*a^2) + (b*tan(e + f*x)^6*(3*a^2 - 10*a*b + 7*b^2) )/(2*a^4))/(f*(a*tan(e + f*x)^5 + b*tan(e + f*x)^7)) - (b^(1/2)*atan((b^(1 /2)*tan(e + f*x)*(a - b)*(3*a - 7*b))/(a^(1/2)*(3*a^2 - 10*a*b + 7*b^2)))* (a - b)*(3*a - 7*b))/(2*a^(9/2)*f)