Integrand size = 25, antiderivative size = 319 \[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a-b) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\sin (e+f x)}{3 (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {(a-b) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{3 a b (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {\operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{3 a (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
-1/3*(a-b)*sin(f*x+e)/b/(a+b)^2/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2 )+1/3*sin(f*x+e)/(a+b)/f/(a+b-a*sin(f*x+e)^2)/(sec(f*x+e)^2*(a+b-a*sin(f*x +e)^2))^(1/2)+1/3*(a-b)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(a+b-a*sin(f *x+e)^2)/a/b/(a+b)^2/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)+1/3*EllipticF(sin(f*x+e),(a/(a+ b))^(1/2))*(1-a*sin(f*x+e)^2/(a+b))^(1/2)/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)
Result contains complex when optimal does not.
Time = 22.18 (sec) , antiderivative size = 1204, normalized size of antiderivative = 3.77 \[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a+2 b+a \cos (2 e+2 f x))^{5/2} \sec ^5(e+f x) \left (-\frac {\left (-2 \sqrt {-\frac {1}{b}} (-a-a \cos (2 e+2 f x)) \left (2 a^2 (a+3 b+a \cos (2 e+2 f x))+b \left (2 b^2+3 b (a+2 b+a \cos (2 e+2 f x))-2 (a+2 b+a \cos (2 e+2 f x))^2\right )+a \left (4 b^2+5 b (a+2 b+a \cos (2 e+2 f x))-(a+2 b+a \cos (2 e+2 f x))^2\right )\right )+2 i \left (a^2+3 a b+2 b^2\right ) \sqrt {\frac {a-a \cos (2 e+2 f x)}{a+b}} (a+2 b+a \cos (2 e+2 f x))^{3/2} \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 \left (2 a^2+5 a b+3 b^2\right ) (a+2 b+a \cos (2 e+2 f x))^{3/2} \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 ) \sin (2 e+2 f x)}{24 a \sqrt {-\frac {1}{b}} b^2 (a+b)^2 f \sqrt {\frac {(a-a \cos (2 e+2 f x)) (a+a \cos (2 e+2 f x))}{a^2}} (a+2 b+a \cos (2 e+2 f x))^{3/2} \sqrt {1-\cos ^2(2 e+2 f x)}}+\frac {\cos (2 (e+f x)) \left (-2 \sqrt {-\frac {1}{b}} (-a-a \cos (2 e+2 f x)) \left (4 b^4-b^2 (a+2 b+a \cos (2 e+2 f x))^2+2 a^3 (a+3 b+a \cos (2 e+2 f x))+a b \left (10 b^2+b (a+2 b+a \cos (2 e+2 f x))-(a+2 b+a \cos (2 e+2 f x))^2\right )+a^2 \left (8 b^2+3 b (a+2 b+a \cos (2 e+2 f x))-(a+2 b+a \cos (2 e+2 f x))^2\right )\right )+2 i \left (a^3+2 a^2 b+2 a b^2+b^3\right ) (a+2 b+a \cos (2 e+2 f x))^{3/2} \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}} 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 \left (2 a^2+3 a b+b^2\right ) (a+2 b+a \cos (2 e+2 f x))^{3/2} \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)}{24 a^2 \sqrt {-\frac {1}{b}} b^2 (a+b)^2 f \sqrt {\frac {(a-a \cos (2 e+2 f x)) (a+a \cos (2 e+2 f x))}{a^2}} (a+2 b+a \cos (2 e+2 f x))^{3/2} \sqrt {1-\cos ^2(2 e+2 f x)}}\right )}{2 \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
((a + 2*b + a*Cos[2*e + 2*f*x])^(5/2)*Sec[e + f*x]^5*(-1/24*((-2*Sqrt[-b^( -1)]*(-a - a*Cos[2*e + 2*f*x])*(2*a^2*(a + 3*b + a*Cos[2*e + 2*f*x]) + b*( 2*b^2 + 3*b*(a + 2*b + a*Cos[2*e + 2*f*x]) - 2*(a + 2*b + a*Cos[2*e + 2*f* x])^2) + a*(4*b^2 + 5*b*(a + 2*b + a*Cos[2*e + 2*f*x]) - (a + 2*b + a*Cos[ 2*e + 2*f*x])^2)) + (2*I)*(a^2 + 3*a*b + 2*b^2)*Sqrt[(a - a*Cos[2*e + 2*f* x])/(a + b)]*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*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*(2*a^2 + 5*a*b + 3*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*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]*EllipticF[I* ArcSinh[(Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*e + 2*f*x]])/Sqrt[2]], b/(a + b)])*Sin[2*e + 2*f*x])/(a*Sqrt[-b^(-1)]*b^2*(a + b)^2*f*Sqrt[((a - a*Cos [2*e + 2*f*x])*(a + a*Cos[2*e + 2*f*x]))/a^2]*(a + 2*b + a*Cos[2*e + 2*f*x ])^(3/2)*Sqrt[1 - Cos[2*e + 2*f*x]^2]) + (Cos[2*(e + f*x)]*(-2*Sqrt[-b^(-1 )]*(-a - a*Cos[2*e + 2*f*x])*(4*b^4 - b^2*(a + 2*b + a*Cos[2*e + 2*f*x])^2 + 2*a^3*(a + 3*b + a*Cos[2*e + 2*f*x]) + a*b*(10*b^2 + b*(a + 2*b + a*Cos [2*e + 2*f*x]) - (a + 2*b + a*Cos[2*e + 2*f*x])^2) + a^2*(8*b^2 + 3*b*(a + 2*b + a*Cos[2*e + 2*f*x]) - (a + 2*b + a*Cos[2*e + 2*f*x])^2)) + (2*I)*(a ^3 + 2*a^2*b + 2*a*b^2 + b^3)*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)*Sqrt[(4 *a + 4*b - 2*(a + 2*b + a*Cos[2*e + 2*f*x]))/(a + b)]*Sqrt[2 - (a + 2*b...
Time = 0.57 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4636, 2057, 2058, 314, 25, 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.
| \(\displaystyle \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (e+f x)^3}{\left (a+b \sec (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^2 \left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^2 \left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\sin ^2(e+f x)}}{\left (-a \sin ^2(e+f x)+a+b\right )^{5/2}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}-\frac {\int -\frac {2-\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{3 (a+b)}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int \frac {2-\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {-\frac {\int -\frac {-\left ((a-b) \sin ^2(e+f x)\right )+a+b}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b (a+b)}-\frac {(a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\int \frac {-\left ((a-b) \sin ^2(e+f x)\right )+a+b}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b (a+b)}-\frac {(a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {b (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}+\frac {(a-b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}}{b (a+b)}-\frac {(a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {(a-b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}+\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {(a-b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}+\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {(a-b) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}+\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{b (a+b)}-\frac {(a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}+\frac {(a-b) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}}{b (a+b)}-\frac {(a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{b (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 (a+b)}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*((Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/( 3*(a + b)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + (-(((a - b)*Sin[e + f*x]*Sqr t[1 - Sin[e + f*x]^2])/(b*(a + b)*Sqrt[a + b - a*Sin[e + f*x]^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[ArcS in[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*(a + b)))/(3*(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)])
3.3.84.3.1 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[((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 = 5.09 (sec) , antiderivative size = 12063, normalized size of antiderivative = 37.82
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 1244, normalized size of antiderivative = 3.90 \[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/6*((2*((-I*a^4 + I*a^3*b)*cos(f*x + e)^4 - I*a^2*b^2 + I*a*b^3 - 2*(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 - I*a^3*b + 2*I*a^2*b^2)*cos(f*x + e)^4 - I*a^2*b^2 - I*a*b^3 + 2*I*b^4 + 2*(-I*a^3*b - I*a^2*b^2 + 2*I*a*b^3)*cos(f*x + e)^2)*sqrt(a))*sqrt((2*a*sq rt((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*((I*a^4 - I*a^3*b) *cos(f*x + e)^4 + I*a^2*b^2 - I*a*b^3 - 2*(-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 + I*a^3*b - 2*I*a^2*b^2)*co s(f*x + e)^4 + I*a^2*b^2 + I*a*b^3 - 2*I*b^4 + 2*(I*a^3*b + 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*(4*(I*a^3*b*cos(f*x + e)^4 + 2*I*a^2*b^2*cos(f* x + e)^2 + I*a*b^3)*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^2 + 4* (a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - 2*(4*(-I*a^3*b*cos(f*x + e)...
\[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sec ^{3}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\cos \left (e+f\,x\right )}^3\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]