Integrand size = 21, antiderivative size = 154 \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{5/2} (a+b)^3 f}-\frac {\text {arctanh}(\cos (e+f x))}{(a+b)^3 f}-\frac {b \cos ^3(e+f x)}{4 a (a+b) f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (7 a+3 b) \cos (e+f x)}{8 a^2 (a+b)^2 f \left (b+a \cos ^2(e+f x)\right )} \]
-arctanh(cos(f*x+e))/(a+b)^3/f-1/4*b*cos(f*x+e)^3/a/(a+b)/f/(b+a*cos(f*x+e )^2)^2-1/8*b*(7*a+3*b)*cos(f*x+e)/a^2/(a+b)^2/f/(b+a*cos(f*x+e)^2)+1/8*(15 *a^2+10*a*b+3*b^2)*arctan(cos(f*x+e)*a^(1/2)/b^(1/2))*b^(1/2)/a^(5/2)/(a+b )^3/f
Result contains complex when optimal does not.
Time = 4.22 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.90 \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^5(e+f x) \left (\frac {8 b^2 (a+b)^2}{a^2}-\frac {2 b (a+b) (9 a+5 b) (a+2 b+a \cos (2 (e+f x)))}{a^2}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2 \sec (e+f x)}{a^{5/2}}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right ) (a+2 b+a \cos (2 (e+f x)))^2 \sec (e+f x)}{a^{5/2}}-8 (a+2 b+a \cos (2 (e+f x)))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right ) \sec (e+f x)+8 (a+2 b+a \cos (2 (e+f x)))^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sec (e+f x)\right )}{64 (a+b)^3 f \left (a+b \sec ^2(e+f x)\right )^3} \]
((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^5*((8*b^2*(a + b)^2)/a^2 - (2 *b*(a + b)*(9*a + 5*b)*(a + 2*b + a*Cos[2*(e + f*x)]))/a^2 + (Sqrt[b]*(15* a^2 + 10*a*b + 3*b^2)*ArcTan[((-Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cos[e] - I*S in[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[e]*(Sqrt[a] - Sqrt[a + b]*Sqrt[(Cos[e ] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2* Sec[e + f*x])/a^(5/2) + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[((-Sqrt[ a] + I*Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2])*Sin[e]*Tan[(f*x)/2] + Cos[ e]*(Sqrt[a] + Sqrt[a + b]*Sqrt[(Cos[e] - I*Sin[e])^2]*Tan[(f*x)/2]))/Sqrt[ b]]*(a + 2*b + a*Cos[2*(e + f*x)])^2*Sec[e + f*x])/a^(5/2) - 8*(a + 2*b + a*Cos[2*(e + f*x)])^2*Log[Cos[(e + f*x)/2]]*Sec[e + f*x] + 8*(a + 2*b + a* Cos[2*(e + f*x)])^2*Log[Sin[(e + f*x)/2]]*Sec[e + f*x]))/(64*(a + b)^3*f*( a + b*Sec[e + f*x]^2)^3)
Time = 0.36 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4621, 372, 440, 397, 218, 219}
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.
| \(\displaystyle \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x) \left (a+b \sec (e+f x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle -\frac {\int \frac {\cos ^6(e+f x)}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^3}d\cos (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle -\frac {\frac {b \cos ^3(e+f x)}{4 a (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\int \frac {\cos ^2(e+f x) \left (3 b-(4 a+3 b) \cos ^2(e+f x)\right )}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )^2}d\cos (e+f x)}{4 a (a+b)}}{f}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle -\frac {\frac {b \cos ^3(e+f x)}{4 a (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\int \frac {b (7 a+3 b)-\left (8 a^2+7 b a+3 b^2\right ) \cos ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) \left (a \cos ^2(e+f x)+b\right )}d\cos (e+f x)}{2 a (a+b)}-\frac {b (7 a+3 b) \cos (e+f x)}{2 a (a+b) \left (a \cos ^2(e+f x)+b\right )}}{4 a (a+b)}}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {b \cos ^3(e+f x)}{4 a (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {b \left (15 a^2+10 a b+3 b^2\right ) \int \frac {1}{a \cos ^2(e+f x)+b}d\cos (e+f x)}{a+b}-\frac {8 a^2 \int \frac {1}{1-\cos ^2(e+f x)}d\cos (e+f x)}{a+b}}{2 a (a+b)}-\frac {b (7 a+3 b) \cos (e+f x)}{2 a (a+b) \left (a \cos ^2(e+f x)+b\right )}}{4 a (a+b)}}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {b \cos ^3(e+f x)}{4 a (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {8 a^2 \int \frac {1}{1-\cos ^2(e+f x)}d\cos (e+f x)}{a+b}}{2 a (a+b)}-\frac {b (7 a+3 b) \cos (e+f x)}{2 a (a+b) \left (a \cos ^2(e+f x)+b\right )}}{4 a (a+b)}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {b \cos ^3(e+f x)}{4 a (a+b) \left (a \cos ^2(e+f x)+b\right )^2}-\frac {\frac {\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}-\frac {8 a^2 \text {arctanh}(\cos (e+f x))}{a+b}}{2 a (a+b)}-\frac {b (7 a+3 b) \cos (e+f x)}{2 a (a+b) \left (a \cos ^2(e+f x)+b\right )}}{4 a (a+b)}}{f}\) |
-(((b*Cos[e + f*x]^3)/(4*a*(a + b)*(b + a*Cos[e + f*x]^2)^2) - (((Sqrt[b]* (15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(Sqrt[a] *(a + b)) - (8*a^2*ArcTanh[Cos[e + f*x]])/(a + b))/(2*a*(a + b)) - (b*(7*a + 3*b)*Cos[e + f*x])/(2*a*(a + b)*(b + a*Cos[e + f*x]^2)))/(4*a*(a + b))) /f)
3.1.57.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && 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[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 1.47 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (-1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}+\frac {b \left (\frac {-\frac {\left (9 a^{2}+14 a b +5 b^{2}\right ) \cos \left (f x +e \right )^{3}}{8 a}-\frac {b \left (7 a^{2}+10 a b +3 b^{2}\right ) \cos \left (f x +e \right )}{8 a^{2}}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (15 a^{2}+10 a b +3 b^{2}\right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a +b \right )^{3}}-\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}}{f}\) | \(156\) |
| default | \(\frac {\frac {\ln \left (-1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}+\frac {b \left (\frac {-\frac {\left (9 a^{2}+14 a b +5 b^{2}\right ) \cos \left (f x +e \right )^{3}}{8 a}-\frac {b \left (7 a^{2}+10 a b +3 b^{2}\right ) \cos \left (f x +e \right )}{8 a^{2}}}{\left (b +a \cos \left (f x +e \right )^{2}\right )^{2}}+\frac {\left (15 a^{2}+10 a b +3 b^{2}\right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a +b \right )^{3}}-\frac {\ln \left (1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}}{f}\) | \(156\) |
| risch | \(-\frac {b \left (9 a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+5 a b \,{\mathrm e}^{7 i \left (f x +e \right )}+27 a^{2} {\mathrm e}^{5 i \left (f x +e \right )}+43 a b \,{\mathrm e}^{5 i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{5 i \left (f x +e \right )}+27 a^{2} {\mathrm e}^{3 i \left (f x +e \right )}+43 a b \,{\mathrm e}^{3 i \left (f x +e \right )}+12 b^{2} {\mathrm e}^{3 i \left (f x +e \right )}+9 a^{2} {\mathrm e}^{i \left (f x +e \right )}+5 a b \,{\mathrm e}^{i \left (f x +e \right )}\right )}{4 a^{2} \left (a +b \right )^{2} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 a \left (a +b \right )^{3} f}+\frac {5 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{8 a^{2} \left (a +b \right )^{3} f}+\frac {3 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b^{2}}{16 a^{3} \left (a +b \right )^{3} f}-\frac {15 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{16 a \left (a +b \right )^{3} f}-\frac {5 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{8 a^{2} \left (a +b \right )^{3} f}-\frac {3 i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b^{2}}{16 a^{3} \left (a +b \right )^{3} f}\) | \(581\) |
1/f*(1/2/(a+b)^3*ln(-1+cos(f*x+e))+b/(a+b)^3*((-1/8*(9*a^2+14*a*b+5*b^2)/a *cos(f*x+e)^3-1/8*b*(7*a^2+10*a*b+3*b^2)/a^2*cos(f*x+e))/(b+a*cos(f*x+e)^2 )^2+1/8*(15*a^2+10*a*b+3*b^2)/a^2/(a*b)^(1/2)*arctan(a*cos(f*x+e)/(a*b)^(1 /2)))-1/2/(a+b)^3*ln(1+cos(f*x+e)))
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (140) = 280\).
Time = 0.40 (sec) , antiderivative size = 779, normalized size of antiderivative = 5.06 \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\left [-\frac {2 \, {\left (9 \, a^{3} b + 14 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left ({\left (15 \, a^{4} + 10 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4} + 2 \, {\left (15 \, a^{3} b + 10 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (7 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4}\right )} \cos \left (f x + e\right ) + 8 \, {\left (a^{4} \cos \left (f x + e\right )^{4} + 2 \, a^{3} b \cos \left (f x + e\right )^{2} + a^{2} b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - 8 \, {\left (a^{4} \cos \left (f x + e\right )^{4} + 2 \, a^{3} b \cos \left (f x + e\right )^{2} + a^{2} b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{16 \, {\left ({\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} f\right )}}, -\frac {{\left (9 \, a^{3} b + 14 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left ({\left (15 \, a^{4} + 10 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4} + 2 \, {\left (15 \, a^{3} b + 10 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) + {\left (7 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4}\right )} \cos \left (f x + e\right ) + 4 \, {\left (a^{4} \cos \left (f x + e\right )^{4} + 2 \, a^{3} b \cos \left (f x + e\right )^{2} + a^{2} b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - 4 \, {\left (a^{4} \cos \left (f x + e\right )^{4} + 2 \, a^{3} b \cos \left (f x + e\right )^{2} + a^{2} b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{8 \, {\left ({\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} f\right )}}\right ] \]
[-1/16*(2*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^3 - ((15*a^4 + 10* a^3*b + 3*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 + 3*b^4 + 2*(15* a^3*b + 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 2*(7*a^ 2*b^2 + 10*a*b^3 + 3*b^4)*cos(f*x + e) + 8*(a^4*cos(f*x + e)^4 + 2*a^3*b*c os(f*x + e)^2 + a^2*b^2)*log(1/2*cos(f*x + e) + 1/2) - 8*(a^4*cos(f*x + e) ^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f), -1/8*((9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^3 - (( 15*a^4 + 10*a^3*b + 3*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 + 3* b^4 + 2*(15*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt(b/a)*arctan (a*sqrt(b/a)*cos(f*x + e)/b) + (7*a^2*b^2 + 10*a*b^3 + 3*b^4)*cos(f*x + e) + 4*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*log(1/2*cos(f *x + e) + 1/2) - 4*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2) *log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*co s(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^ 2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f)]
Timed out. \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.69 \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {{\left (15 \, a^{2} b + 10 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {a b}} - \frac {{\left (9 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (7 \, a b^{2} + 3 \, b^{3}\right )} \cos \left (f x + e\right )}{a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} \cos \left (f x + e\right )^{2}} - \frac {4 \, \log \left (\cos \left (f x + e\right ) + 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {4 \, \log \left (\cos \left (f x + e\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}{8 \, f} \]
1/8*((15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(a*cos(f*x + e)/sqrt(a*b))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)) - ((9*a^2*b + 5*a*b^2)*cos(f*x + e)^3 + (7*a*b^2 + 3*b^3)*cos(f*x + e))/(a^4*b^2 + 2*a^3*b^3 + a^2*b^4 + (a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^4 + 2*(a^5*b + 2*a^4*b^2 + a^3*b^3 )*cos(f*x + e)^2) - 4*log(cos(f*x + e) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 ) + 4*log(cos(f*x + e) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3))/f
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (140) = 280\).
Time = 0.41 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.84 \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {{\left (15 \, a^{2} b + 10 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b}{\sqrt {a b} \cos \left (f x + e\right ) + \sqrt {a b}}\right )}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {a b}} - \frac {4 \, \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (9 \, a^{3} b + 21 \, a^{2} b^{2} + 15 \, a b^{3} + 3 \, b^{4} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {13 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {23 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {9 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {9 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {21 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {13 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {3 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a + b + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{2}}}{8 \, f} \]
-1/8*((15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(-(a*cos(f*x + e) - b)/(sqrt(a*b )*cos(f*x + e) + sqrt(a*b)))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a *b)) - 4*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 2*(9*a^3*b + 21*a^2*b^2 + 15*a*b^3 + 3*b^4 + 27*a^3*b*( cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 13*a^2*b^2*(cos(f*x + e) - 1)/(cos( f*x + e) + 1) - 23*a*b^3*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 9*b^4*(co s(f*x + e) - 1)/(cos(f*x + e) + 1) + 27*a^3*b*(cos(f*x + e) - 1)^2/(cos(f* x + e) + 1)^2 - 9*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 21*a *b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 9*b^4*(cos(f*x + e) - 1)^ 2/(cos(f*x + e) + 1)^2 + 9*a^3*b*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 13*a*b^3*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 3*b^4*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*(a + b + 2*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 2*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + a*( cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)^2))/f
Time = 21.90 (sec) , antiderivative size = 3557, normalized size of antiderivative = 23.10 \[ \int \frac {\csc (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
(atan((((-a^5*b)^(1/2)*((cos(e + f*x)*(60*a*b^5 + 64*a^6 + 9*b^6 + 190*a^2 *b^4 + 300*a^3*b^3 + 225*a^4*b^2))/(32*(4*a^6*b + a^7 + a^3*b^4 + 4*a^4*b^ 3 + 6*a^5*b^2)) + (((224*a^10*b + 96*a^3*b^8 + 800*a^4*b^7 + 2784*a^5*b^6 + 5280*a^6*b^5 + 5920*a^7*b^4 + 3936*a^8*b^3 + 1440*a^9*b^2)/(64*(6*a^8*b + a^9 + a^3*b^6 + 6*a^4*b^5 + 15*a^5*b^4 + 20*a^6*b^3 + 15*a^7*b^2)) - (co s(e + f*x)*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2)*(1280*a^11*b + 256*a^1 2 - 256*a^5*b^7 - 1280*a^6*b^6 - 2304*a^7*b^5 - 1280*a^8*b^4 + 1280*a^9*b^ 3 + 2304*a^10*b^2))/(512*(3*a^7*b + a^8 + a^5*b^3 + 3*a^6*b^2)*(4*a^6*b + a^7 + a^3*b^4 + 4*a^4*b^3 + 6*a^5*b^2)))*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2))/(16*(3*a^7*b + a^8 + a^5*b^3 + 3*a^6*b^2)))*(10*a*b + 15*a^2 + 3* b^2)*1i)/(16*(3*a^7*b + a^8 + a^5*b^3 + 3*a^6*b^2)) + ((-a^5*b)^(1/2)*((co s(e + f*x)*(60*a*b^5 + 64*a^6 + 9*b^6 + 190*a^2*b^4 + 300*a^3*b^3 + 225*a^ 4*b^2))/(32*(4*a^6*b + a^7 + a^3*b^4 + 4*a^4*b^3 + 6*a^5*b^2)) - (((224*a^ 10*b + 96*a^3*b^8 + 800*a^4*b^7 + 2784*a^5*b^6 + 5280*a^6*b^5 + 5920*a^7*b ^4 + 3936*a^8*b^3 + 1440*a^9*b^2)/(64*(6*a^8*b + a^9 + a^3*b^6 + 6*a^4*b^5 + 15*a^5*b^4 + 20*a^6*b^3 + 15*a^7*b^2)) + (cos(e + f*x)*(-a^5*b)^(1/2)*( 10*a*b + 15*a^2 + 3*b^2)*(1280*a^11*b + 256*a^12 - 256*a^5*b^7 - 1280*a^6* b^6 - 2304*a^7*b^5 - 1280*a^8*b^4 + 1280*a^9*b^3 + 2304*a^10*b^2))/(512*(3 *a^7*b + a^8 + a^5*b^3 + 3*a^6*b^2)*(4*a^6*b + a^7 + a^3*b^4 + 4*a^4*b^3 + 6*a^5*b^2)))*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2))/(16*(3*a^7*b + ...