Integrand size = 25, antiderivative size = 289 \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\frac {a (2 a+b) \sin (e+f x)}{b^2 (a+b) f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {(2 a+b) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{b^2 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {\operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{b f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\sec (e+f x) \tan (e+f x)}{b f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
a*(2*a+b)*sin(f*x+e)/b^2/(a+b)/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2) -(2*a+b)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(a+b-a*sin(f*x+e)^2)/b^2/(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)/b/f/(cos(f*x+e)^2)^(1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x +e)^2))^(1/2)+sec(f*x+e)*tan(f*x+e)/b/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2) )^(1/2)
\[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx \]
Time = 0.57 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 4636, 2057, 2058, 316, 25, 27, 402, 25, 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.
\(\displaystyle \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (e+f x)^5}{\left (a+b \sec (e+f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4636 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^3 \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{3/2}}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^3 \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{3/2}}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {1}{\left (1-\sin ^2(e+f x)\right )^{3/2} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 316 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int -\frac {a \left (\sin ^2(e+f x)+1\right )}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{b}+\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {\int \frac {a \left (\sin ^2(e+f x)+1\right )}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \int \frac {\sin ^2(e+f x)+1}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (-\frac {\int -\frac {2 (a+b)-(2 a+b) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b (a+b)}-\frac {(2 a+b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (\frac {\int \frac {2 (a+b)-(2 a+b) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b (a+b)}-\frac {(2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (\frac {\frac {(2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}}{b (a+b)}-\frac {(2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (\frac {\frac {(2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (\frac {\frac {(2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 330 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (\frac {\frac {(2 a+b) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x)}{b \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}-\frac {a \left (\frac {\frac {(2 a+b) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(2 a+b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*(Sin[e + f*x]/(b*Sqrt[1 - Sin[e + f*x]^2]* Sqrt[a + b - a*Sin[e + f*x]^2]) - (a*(-(((2*a + b)*Sin[e + f*x]*Sqrt[1 - S in[e + f*x]^2])/(b*(a + b)*Sqrt[a + b - a*Sin[e + f*x]^2])) + (((2*a + b)* 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)]) - (b*(a + b)*EllipticF[ArcSin[Si n[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]))/(b*(a + b))))/b))/(f*Sqrt[1 - Sin[e + f*x]^2]*Sqrt [(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x]^2)])
3.3.70.3.1 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[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[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 = 9.40 (sec) , antiderivative size = 10329, normalized size of antiderivative = 35.74
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 960, normalized size of antiderivative = 3.32 \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
1/2*((2*(2*I*a^2*b + I*a*b^2 + (2*I*a^3 + I*a^2*b)*cos(f*x + e)^2)*sqrt(a) *sqrt((a*b + b^2)/a^2) - (2*I*a^2*b + 5*I*a*b^2 + 2*I*b^3 + (2*I*a^3 + 5*I *a^2*b + 2*I*a*b^2)*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^2*b - I*a*b^2 + (-2*I*a^3 - I*a^2*b)*cos(f*x + e)^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) - (-2*I*a^2*b - 5* I*a*b^2 - 2*I*b^3 + (-2*I*a^3 - 5*I*a^2*b - 2*I*a*b^2)*cos(f*x + e)^2)*sqr t(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) - 4*( (-I*a^2*b*cos(f*x + e)^2 - I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) + (-I*a^ 2*b - 3*I*a*b^2 - 2*I*b^3 + (-I*a^3 - 3*I*a^2*b - 2*I*a*b^2)*cos(f*x + e)^ 2)*sqrt(a))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_f(arcsi n(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*b*cos(f*x + e)^2 + I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2) + ( I*a^2*b + 3*I*a*b^2 + 2*I*b^3 + (I*a^3 + 3*I*a^2*b + 2*I*a*b^2)*cos(f*x + e)^2)*sqrt(a))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_f(ar csin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*si...
\[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sec ^{5}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sec \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\sec \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\cos \left (e+f\,x\right )}^5\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]