Integrand size = 23, antiderivative size = 327 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 (2 a+b) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {2 (2 a+b) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{3 a^2 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {(3 a+2 b) \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{3 a^2 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
2/3*(2*a+b)*sin(f*x+e)/a/(a+b)^2/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/ 2)-1/3*b*sin(f*x+e)/a/(a+b)/f/(a+b-a*sin(f*x+e)^2)/(sec(f*x+e)^2*(a+b-a*si n(f*x+e)^2))^(1/2)-2/3*(2*a+b)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(a+b- a*sin(f*x+e)^2)/a^2/(a+b)^2/f/(cos(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*si n(f*x+e)^2))^(1/2)/(1-a*sin(f*x+e)^2/(a+b))^(1/2)+1/3*(3*a+2*b)*EllipticF( sin(f*x+e),(a/(a+b))^(1/2))*(1-a*sin(f*x+e)^2/(a+b))^(1/2)/a^2/(a+b)/f/(co s(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Time = 0.58 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 4636, 2057, 2058, 315, 25, 402, 27, 399, 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.
| \(\displaystyle \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (e+f x)}{\left (a+b \sec (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right ) \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right ) \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\left (1-\sin ^2(e+f x)\right )^{3/2}}{\left (-a \sin ^2(e+f x)+a+b\right )^{5/2}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (-\frac {\int -\frac {-\left ((3 a+2 b) \sin ^2(e+f x)\right )+3 a+b}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}-\frac {b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int \frac {-\left ((3 a+2 b) \sin ^2(e+f x)\right )+3 a+b}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\int \frac {b \left (-2 (2 a+b) \sin ^2(e+f x)+a+b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b (a+b)}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\int \frac {-2 (2 a+b) \sin ^2(e+f x)+a+b}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a+b}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\frac {2 (2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {(a+b) (3 a+2 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}}{a+b}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\frac {2 (2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {(a+b) (3 a+2 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{a+b}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\frac {2 (2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {(a+b) (3 a+2 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{a+b}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\frac {2 (2 a+b) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a+b) (3 a+2 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{a+b}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\frac {2 (2 a+b) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a+b) (3 a+2 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{a+b}}{3 a (a+b)}-\frac {b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*(-1/3*(b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x ]^2])/(a*(a + b)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + ((2*(2*a + b)*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/((a + b)*Sqrt[a + b - a*Sin[e + f*x]^2]) - ((2*(2*a + b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*S in[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a + b)*(3*a + 2*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))/(3*a*(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)])
3.3.85.3.1 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)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[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[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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 = 6.02 (sec) , antiderivative size = 16966, normalized size of antiderivative = 51.88
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.95 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/3*((2*((2*I*a^4 + I*a^3*b)*cos(f*x + e)^4 + 2*I*a^2*b^2 + I*a*b^3 - 2*(- 2*I*a^3*b - I*a^2*b^2)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - ((2 *I*a^4 + 5*I*a^3*b + 2*I*a^2*b^2)*cos(f*x + e)^4 + 2*I*a^2*b^2 + 5*I*a*b^3 + 2*I*b^4 + 2*(2*I*a^3*b + 5*I*a^2*b^2 + 2*I*a*b^3)*cos(f*x + e)^2)*sqrt( a))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_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*(( -2*I*a^4 - I*a^3*b)*cos(f*x + e)^4 - 2*I*a^2*b^2 - I*a*b^3 - 2*(2*I*a^3*b + I*a^2*b^2)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - ((-2*I*a^4 - 5*I*a^3*b - 2*I*a^2*b^2)*cos(f*x + e)^4 - 2*I*a^2*b^2 - 5*I*a*b^3 - 2*I*b^ 4 + 2*(-2*I*a^3*b - 5*I*a^2*b^2 - 2*I*a*b^3)*cos(f*x + e)^2)*sqrt(a))*sqrt ((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_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*((-3*I*a^4 - I*a^3*b)*cos(f*x + e)^4 - 3*I*a^2*b^2 - I*a*b^3 - 2*(3*I*a^3*b + I*a^2* b^2)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - ((-I*a^4 - 3*I*a^3*b - 2*I*a^2*b^2)*cos(f*x + e)^4 - I*a^2*b^2 - 3*I*a*b^3 - 2*I*b^4 + 2*(-I*a^ 3*b - 3*I*a^2*b^2 - 2*I*a*b^3)*cos(f*x + e)^2)*sqrt(a))*sqrt((2*a*sqrt((a* b + b^2)/a^2) - a - 2*b)/a)*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^...
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]