Integrand size = 25, antiderivative size = 372 \[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=-\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b^2 f}+\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b^2 f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {(a-8 b) (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{15 b f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {(a+4 b) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{15 b f}+\frac {\sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{5 f} \]
-1/15*(2*a^2-3*a*b-8*b^2)*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^( 1/2)/b^2/f+1/15*(2*a^2-3*a*b-8*b^2)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))* (cos(f*x+e)^2)^(1/2)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/b^2/f/(1-a* sin(f*x+e)^2/(a+b))^(1/2)-1/15*(a-8*b)*(a+b)*EllipticF(sin(f*x+e),(a/(a+b) )^(1/2))*(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)/b/f/(a+b-a*sin(f*x+e)^2)+1/15*(a+4*b)*sec(f*x +e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/b/f+1/5*sec(f*x+e )^3*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f
\[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx \]
Time = 0.66 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4636, 2057, 2058, 314, 25, 402, 25, 402, 25, 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.
| \(\displaystyle \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (e+f x)^5 \sqrt {a+b \sec (e+f x)^2}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+\frac {b}{1-\sin ^2(e+f x)}}}{\left (1-\sin ^2(e+f x)\right )^3}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {\sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}{\left (1-\sin ^2(e+f x)\right )^3}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\left (1-\sin ^2(e+f x)\right )^{7/2}}d\sin (e+f x)}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}-\frac {1}{5} \int -\frac {4 (a+b)-3 a \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \int \frac {4 (a+b)-3 a \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {\int -\frac {a (a+4 b) \sin ^2(e+f x)+(a-8 b) (a+b)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 b}+\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {a (a+4 b) \sin ^2(e+f x)+(a-8 b) (a+b)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int -\frac {a \left (2 (a-2 b) (a+b)-\left (2 a^2-3 b a-8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}+\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {\int \frac {a \left (2 (a-2 b) (a+b)-\left (2 a^2-3 b a-8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \int \frac {2 (a-2 b) (a+b)-\left (2 a^2-3 b a-8 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a-8 b) (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {(a+4 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 b \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\left (2 a^2-3 a b-8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (2 a^2-3 a b-8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a-8 b) (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}}{3 b}\right )+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
(Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x ]^2)]*((Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*(1 - Sin[e + f*x]^ 2)^(5/2)) + (((a + 4*b)*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(3*b* (1 - Sin[e + f*x]^2)^(3/2)) - (((2*a^2 - 3*a*b - 8*b^2)*Sin[e + f*x]*Sqrt[ a + b - a*Sin[e + f*x]^2])/(b*Sqrt[1 - Sin[e + f*x]^2]) - (a*(((2*a^2 - 3* a*b - 8*b^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 - 8*b)*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])))/b)/(3*b))/5))/(f*Sqrt[a + b - a*Sin[e + f*x]^2])
3.3.28.3.1 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[(-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[((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 = 14.83 (sec) , antiderivative size = 8814, normalized size of antiderivative = 23.69
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.38 \[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {{\left (2 \, {\left (-2 i \, a^{3} + 3 i \, a^{2} b + 8 i \, a b^{2}\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} \cos \left (f x + e\right )^{4} - {\left (-2 i \, a^{3} - i \, a^{2} b + 14 i \, a b^{2} + 16 i \, b^{3}\right )} \sqrt {a} \cos \left (f x + e\right )^{4}\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 (2 i \, a^{3} - 3 i \, a^{2} b - 8 i \, a b^{2}\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} \cos \left (f x + e\right )^{4} - {\left (2 i \, a^{3} + i \, a^{2} b - 14 i \, a b^{2} - 16 i \, b^{3}\right )} \sqrt {a} \cos \left (f x + e\right )^{4}\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}}) - 4 \, {\left ({\left (i \, a^{2} b + 4 i \, a b^{2}\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} \cos \left (f x + e\right )^{4} + {\left (i \, a^{3} + i \, a^{2} b - 4 i \, a b^{2} - 4 i \, b^{3}\right )} \sqrt {a} \cos \left (f x + e\right )^{4}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} 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 ({\left (-i \, a^{2} b - 4 i \, a b^{2}\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} \cos \left (f x + e\right )^{4} + {\left (-i \, a^{3} - i \, a^{2} b + 4 i \, a b^{2} + 4 i \, b^{3}\right )} \sqrt {a} \cos \left (f x + e\right )^{4}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} 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}}) - 2 \, {\left ({\left (2 \, a^{3} - 3 \, a^{2} b - 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} - 3 \, a b^{2} - {\left (a^{2} b + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{30 \, a b^{2} f \cos \left (f x + e\right )^{4}} \]
1/30*((2*(-2*I*a^3 + 3*I*a^2*b + 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)* cos(f*x + e)^4 - (-2*I*a^3 - I*a^2*b + 14*I*a*b^2 + 16*I*b^3)*sqrt(a)*cos( f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(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) + (2*(2*I*a^3 - 3*I*a^2*b - 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos( f*x + e)^4 - (2*I*a^3 + I*a^2*b - 14*I*a*b^2 - 16*I*b^3)*sqrt(a)*cos(f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin(sqr t((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 + 4*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + (I*a ^3 + I*a^2*b - 4*I*a*b^2 - 4*I*b^3)*sqrt(a)*cos(f*x + e)^4)*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 ^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - 4*((-I*a^2*b - 4*I*a*b^ 2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + (-I*a^3 - I*a^2*b + 4*I* a*b^2 + 4*I*b^3)*sqrt(a)*cos(f*x + e)^4)*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^2 + 4*(a^2 + 2*a*b )*sqrt((a*b + b^2)/a^2))/a^2) - 2*((2*a^3 - 3*a^2*b - 8*a*b^2)*cos(f*x ...
\[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \sqrt {a + b \sec ^{2}{\left (e + f x \right )}} \sec ^{5}{\left (e + f x \right )}\, dx \]
\[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{5} \,d x } \]
\[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{5} \,d x } \]
Timed out. \[ \int \sec ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \frac {\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}{{\cos \left (e+f\,x\right )}^5} \,d x \]