Integrand size = 23, antiderivative size = 229 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sin (e+f x)}{(a+b) f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{a (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {\operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{a f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
sin(f*x+e)/(a+b)/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)-EllipticE(sin (f*x+e),(a/(a+b))^(1/2))*(a+b-a*sin(f*x+e)^2)/a/(a+b)/f/(cos(f*x+e)^2)^(1/ 2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/(1-a*sin(f*x+e)^2/(a+b))^(1/2 )+EllipticF(sin(f*x+e),(a/(a+b))^(1/2))*(1-a*sin(f*x+e)^2/(a+b))^(1/2)/a/f /(cos(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)
Result contains complex when optimal does not.
Time = 18.39 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.95 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {(a+2 b+a \cos (2 e+2 f x))^{3/2} \sec ^3(e+f x) \left (\frac {\sqrt {-\frac {1}{b}} \left (\left (-\frac {1}{b}\right )^{3/2} b (-a+a \cos (2 e+2 f x)) (a+a \cos (2 e+2 f x))+i (a+b) \sqrt {\frac {a-a \cos (2 e+2 f x)}{a+b}} \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {4-\frac {2 (a+2 b+a \cos (2 e+2 f x))}{b}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 e+2 f x)}}{\sqrt {2}}\right )|\frac {b}{a+b}\right )-i (a+b) \sqrt {\frac {a-a \cos (2 e+2 f x)}{a+b}} \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {4-\frac {2 (a+2 b+a \cos (2 e+2 f x))}{b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 e+2 f x)}}{\sqrt {2}}\right ),\frac {b}{a+b}\right )\right ) \sin (2 e+2 f x)}{4 a (a+b) f \sqrt {\frac {(a-a \cos (2 e+2 f x)) (a+a \cos (2 e+2 f x))}{a^2}} \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {1-\cos ^2(2 e+2 f x)}}-\frac {\sqrt {-\frac {1}{b}} \cos (2 (e+f x)) \left (\left (-\frac {1}{b}\right )^{3/2} b (a+2 b) (-a+a \cos (2 e+2 f x)) (a+a \cos (2 e+2 f x))+i \left (a^2+3 a b+2 b^2\right ) \sqrt {\frac {a-a \cos (2 e+2 f x)}{a+b}} \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {4-\frac {2 (a+2 b+a \cos (2 e+2 f x))}{b}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 e+2 f x)}}{\sqrt {2}}\right )|\frac {b}{a+b}\right )-i a (a+b) \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {\frac {4 a+4 b-2 (a+2 b+a \cos (2 e+2 f x))}{a+b}} \sqrt {2-\frac {a+2 b+a \cos (2 e+2 f x)}{b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 e+2 f x)}}{\sqrt {2}}\right ),\frac {b}{a+b}\right )\right ) \sec \left (2 \left (e+\frac {1}{2} (-2 e+\arccos (\cos (2 e+2 f x)))\right )\right ) \sin (2 e+2 f x)}{4 a^2 (a+b) f \sqrt {\frac {(a-a \cos (2 e+2 f x)) (a+a \cos (2 e+2 f x))}{a^2}} \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {1-\cos ^2(2 e+2 f x)}}\right )}{2 \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
((a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sec[e + f*x]^3*((Sqrt[-b^(-1)]*((-b^ (-1))^(3/2)*b*(-a + a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]) + I*(a + b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x ]]*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticE[I*ArcSinh[(Sqr t[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], b/(a + b)] - I*(a + b)*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*Sqrt[a + 2*b + a*Cos[2*e + 2* f*x]]*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticF[I*ArcSinh[( Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], b/(a + b)])*Si n[2*e + 2*f*x])/(4*a*(a + b)*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2 *e + 2*f*x]))/a^2]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sqrt[1 - Cos[2*e + 2 *f*x]^2]) - (Sqrt[-b^(-1)]*Cos[2*(e + f*x)]*((-b^(-1))^(3/2)*b*(a + 2*b)*( -a + a*Cos[2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]) + I*(a^2 + 3*a*b + 2*b^2 )*Sqrt[(a - a*Cos[2*e + 2*f*x])/(a + b)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x] ]*Sqrt[4 - (2*(a + 2*b + a*Cos[2*e + 2*f*x]))/b]*EllipticE[I*ArcSinh[(Sqrt [-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], b/(a + b)] - I*a*( a + b)*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]]*Sqrt[(4*a + 4*b - 2*(a + 2*b + a *Cos[2*e + 2*f*x]))/(a + b)]*Sqrt[2 - (a + 2*b + a*Cos[2*e + 2*f*x])/b]*El lipticF[I*ArcSinh[(Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[ 2]], b/(a + b)])*Sec[2*(e + (-2*e + ArcCos[Cos[2*e + 2*f*x]])/2)]*Sin[2*e + 2*f*x])/(4*a^2*(a + b)*f*Sqrt[((a - a*Cos[2*e + 2*f*x])*(a + a*Cos[2*...
Time = 0.49 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4636, 2057, 2058, 314, 25, 389, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (e+f x)}{\left (a+b \sec (e+f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right ) \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{3/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right ) \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{3/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\sin ^2(e+f x)}}{\left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\int -\frac {\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a+b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int \frac {\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a+b}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 389 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {(a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}-\frac {\int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}}{a+b}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {(a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}}{a+b}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {(a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}}{a+b}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {(a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}}{a+b}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {(a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}}{a+b}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*((Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/( (a + b)*Sqrt[a + b - a*Sin[e + f*x]^2]) + (-((EllipticE[ArcSin[Sin[e + f*x ]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x] ^2)/(a + b)])) + ((a + b)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[ 1 - (a*Sin[e + f*x]^2)/(a + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2]))/(a + b)))/(f*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[ e + f*x]^2)])
3.3.72.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* (p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d *(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[1/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b Int [1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N eQ[b*c - a*d, 0] && !SimplerSqrtQ[-b/a, -d/c]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x , Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 5.98 (sec) , antiderivative size = 2534, normalized size of antiderivative = 11.07
1/f/(2*I*a^(1/2)*b^(1/2)-a+b)/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/(a+b )/a*(-2*I*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a^(3/2)*b^(1/2)*(csc(f*x +e)-cot(f*x+e))-2*I*a^(1/2)*b^(3/2)*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^ 2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+ e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2 -a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b ))^(1/2)*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot (f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2 )^(1/2))-2*I*a^(3/2)*b^(1/2)*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f *x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a- b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-c os(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2 )*EllipticF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e) ),(-(4*I*a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2) )+2*I*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a^(1/2)*b^(3/2)*(1-cos(f*x+e ))^3*csc(f*x+e)^3+2*I*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*a^(3/2)*b^(1 /2)*(1-cos(f*x+e))^3*csc(f*x+e)^3+2*I*a^(1/2)*b^(3/2)*((2*I*a^(1/2)*b^(1/2 )+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e))-((2*I*a^(1/2)*b^(1/2)+a-b)/(a+ b))^(1/2)*a^2*(1-cos(f*x+e))^3*csc(f*x+e)^3+((2*I*a^(1/2)*b^(1/2)+a-b)/(a+ b))^(1/2)*b^2*(1-cos(f*x+e))^3*csc(f*x+e)^3+2*(-(2*I*a^(1/2)*b^(1/2)*(1...
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.33 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {2 \, a^{2} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - 4 \, {\left (i \, a^{2} \cos \left (f x + e\right )^{2} + i \, a b\right )} \sqrt {a} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} \sqrt {\frac {a b + b^{2}}{a^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) - 4 \, {\left (-i \, a^{2} \cos \left (f x + e\right )^{2} - i \, a b\right )} \sqrt {a} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} \sqrt {\frac {a b + b^{2}}{a^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + {\left (2 \, {\left (i \, a^{2} \cos \left (f x + e\right )^{2} + i \, a b\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} - {\left ({\left (i \, a^{2} + 2 i \, a b\right )} \cos \left (f x + e\right )^{2} + i \, a b + 2 i \, b^{2}\right )} \sqrt {a}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} E(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + {\left (2 \, {\left (-i \, a^{2} \cos \left (f x + e\right )^{2} - i \, a b\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} - {\left ({\left (-i \, a^{2} - 2 i \, a b\right )} \cos \left (f x + e\right )^{2} - i \, a b - 2 i \, b^{2}\right )} \sqrt {a}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} E(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}})}{2 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}} \]
1/2*(2*a^2*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2*sin( f*x + e) - 4*(I*a^2*cos(f*x + e)^2 + I*a*b)*sqrt(a)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*sqrt((a*b + b^2)/a^2)*elliptic_f(arcsin(sqrt((2*a* sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - 4*(-I*a^2 *cos(f*x + e)^2 - I*a*b)*sqrt(a)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b )/a)*sqrt((a*b + b^2)/a^2)*elliptic_f(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^ 2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + (2*(I*a^2*cos(f*x + e)^2 + I *a*b)*sqrt(a)*sqrt((a*b + b^2)/a^2) - ((I*a^2 + 2*I*a*b)*cos(f*x + e)^2 + I*a*b + 2*I*b^2)*sqrt(a))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*el liptic_e(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e ) + I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b ^2)/a^2))/a^2) + (2*(-I*a^2*cos(f*x + e)^2 - I*a*b)*sqrt(a)*sqrt((a*b + b^ 2)/a^2) - ((-I*a^2 - 2*I*a*b)*cos(f*x + e)^2 - I*a*b - 2*I*b^2)*sqrt(a))*s qrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin(sqrt((2*a*s qrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2))/((a^4 + a^3 *b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sec \left (f x + e\right )}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sec \left (f x + e\right )}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]