Integrand size = 25, antiderivative size = 371 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b f}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{15 b f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {(a+b) (9 a+8 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{15 f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {2 (3 a+2 b) \sec (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{15 f}+\frac {b \sec ^3(e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \tan (e+f x)}{5 f} \]
1/15*(3*a^2+13*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/f-1/15*(3*a^2+13*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/f/(1-a*sin (f*x+e)^2/(a+b))^(1/2)+1/15*(a+b)*(9*a+8*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)/f/(a+b-a*sin(f*x+e)^2)+2/15*(3*a+2*b)*sec(f*x+ e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*tan(f*x+e)/f+1/5*b*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
\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx \]
Time = 0.67 (sec) , antiderivative size = 389, 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, 315, 25, 402, 27, 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.
| \(\displaystyle \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (e+f x)^3 \left (a+b \sec (e+f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {\left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{3/2}}{\left (1-\sin ^2(e+f x)\right )^2}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {\left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{3/2}}{\left (1-\sin ^2(e+f x)\right )^2}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \frac {\left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}{\left (1-\sin ^2(e+f x)\right )^{7/2}}d\sin (e+f x)}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}-\frac {1}{5} \int -\frac {(a+b) (5 a+4 b)-a (5 a+3 b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \int \frac {(a+b) (5 a+4 b)-a (5 a+3 b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {\int \frac {b \left ((a+b) (9 a+8 b)-2 a (3 a+2 b) \sin ^2(e+f x)\right )}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 b}+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b) (9 a+8 b)-2 a (3 a+2 b) \sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int -\frac {a \left ((a+b) (3 a+4 b)-\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}+\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {\int \frac {a \left ((a+b) (3 a+4 b)-\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \int \frac {(a+b) (3 a+4 b)-\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) (9 a+8 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) (9 a+8 b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{b}\right )+\frac {2 (3 a+2 b) \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 \left (1-\sin ^2(e+f x)\right )^{3/2}}\right )+\frac {b \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{5 \left (1-\sin ^2(e+f x)\right )^{5/2}}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
(Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x ]^2)]*((b*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*(1 - Sin[e + f*x ]^2)^(5/2)) + ((2*(3*a + 2*b)*Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x]^2]) /(3*(1 - Sin[e + f*x]^2)^(3/2)) + (((3*a^2 + 13*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*(((3*a^2 + 13*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)]) - (b*(a + b)* (9*a + 8*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)/5))/(f*Sqrt[ a + b - a*Sin[e + f*x]^2])
3.3.42.3.1 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 = 13.87 (sec) , antiderivative size = 8946, normalized size of antiderivative = 24.11
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.40 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\frac {{\left (2 \, {\left (3 i \, a^{3} + 13 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 (3 i \, a^{3} + 19 i \, a^{2} b + 34 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 (-3 i \, a^{3} - 13 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 (-3 i \, a^{3} - 19 i \, a^{2} b - 34 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}}) - 2 \, {\left (4 \, {\left (3 i \, a^{2} b + 2 i \, a b^{2}\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} \cos \left (f x + e\right )^{4} + {\left (-3 i \, a^{3} - 13 i \, a^{2} b - 18 i \, a b^{2} - 8 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 (4 \, {\left (-3 i \, a^{2} b - 2 i \, a b^{2}\right )} \sqrt {a} \sqrt {\frac {a b + b^{2}}{a^{2}}} \cos \left (f x + e\right )^{4} + {\left (3 i \, a^{3} + 13 i \, a^{2} b + 18 i \, a b^{2} + 8 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 (3 \, a^{3} + 13 \, a^{2} b + 8 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \, a b^{2} + 2 \, {\left (3 \, a^{2} b + 2 \, 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 f \cos \left (f x + e\right )^{4}} \]
1/30*((2*(3*I*a^3 + 13*I*a^2*b + 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)* cos(f*x + e)^4 - (3*I*a^3 + 19*I*a^2*b + 34*I*a*b^2 + 16*I*b^3)*sqrt(a)*co s(f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arc sin(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^3 - 13*I*a^2*b - 8*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)* cos(f*x + e)^4 - (-3*I*a^3 - 19*I*a^2*b - 34*I*a*b^2 - 16*I*b^3)*sqrt(a)*c os(f*x + e)^4)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(ar csin(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*(4*(3*I*a^2*b + 2*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + (-3*I*a^3 - 13*I*a^2*b - 18*I*a*b^2 - 8*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*(4 *(-3*I*a^2*b - 2*I*a*b^2)*sqrt(a)*sqrt((a*b + b^2)/a^2)*cos(f*x + e)^4 + ( 3*I*a^3 + 13*I*a^2*b + 18*I*a*b^2 + 8*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*((3*a^3 +...
\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \sec ^{3}{\left (e + f x \right )}\, dx \]
\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{3} \,d x } \]
\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^3} \,d x \]