Integrand size = 25, antiderivative size = 55 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=\frac {2 \sqrt {-2-3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),6\right ) \sqrt {\sec (c+d x)}}{d \sqrt {-3-2 \sec (c+d x)}} \] Output:
2*(-2-3*cos(d*x+c))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*Pi+1/2*c,6^(1/2))*se c(d*x+c)^(1/2)/d/(-3-2*sec(d*x+c))^(1/2)
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=\frac {2 \sqrt {2+3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-3-2 \sec (c+d x)}} \] Input:
Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-3 - 2*Sec[c + d*x]],x]
Output:
(2*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]] )/(Sqrt[5]*d*Sqrt[-3 - 2*Sec[c + d*x]])
Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 4345, 3042, 3141}
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.
\(\displaystyle \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2 \sec (c+d x)-3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-2 \csc \left (c+d x+\frac {\pi }{2}\right )-3}}dx\) |
\(\Big \downarrow \) 4345 |
\(\displaystyle \frac {\sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-3 \cos (c+d x)-2}}dx}{\sqrt {-2 \sec (c+d x)-3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}}dx}{\sqrt {-2 \sec (c+d x)-3}}\) |
\(\Big \downarrow \) 3141 |
\(\displaystyle \frac {2 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),6\right )}{d \sqrt {-2 \sec (c+d x)-3}}\) |
Input:
Int[Sqrt[Sec[c + d*x]]/Sqrt[-3 - 2*Sec[c + d*x]],x]
Output:
(2*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6]*Sqrt[Sec[c + d *x]])/(d*Sqrt[-3 - 2*Sec[c + d*x]])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a - b]))*EllipticF[(1/2)*(c + Pi/2 + d*x), -2*(b/(a - b))], x] /; FreeQ [{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 1.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.18
method | result | size |
default | \(\frac {i \sqrt {2}\, \sqrt {-3-2 \sec \left (d x +c \right )}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {\sec \left (d x +c \right )}\, \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right )}{5 d \left (2+3 \cos \left (d x +c \right )\right )}\) | \(120\) |
Input:
int(sec(d*x+c)^(1/2)/(-3-2*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5*I/d*2^(1/2)*(-3-2*sec(d*x+c))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*10^(1/2)* ((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(csc(d*x+c)-cot(d*x+c) ),1/5*I*5^(1/2))*sec(d*x+c)^(1/2)/(2+3*cos(d*x+c))*(cos(d*x+c)^2+cos(d*x+c ))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=-\frac {\sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )}{3 \, d} \] Input:
integrate(sec(d*x+c)^(1/2)/(-3-2*sec(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-1/3*(sqrt(6)*weierstrassPInverse(-44/27, 784/729, cos(d*x + c) + I*sin(d* x + c) + 4/9) + sqrt(6)*weierstrassPInverse(-44/27, 784/729, cos(d*x + c) - I*sin(d*x + c) + 4/9))/d
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=\int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {- 2 \sec {\left (c + d x \right )} - 3}}\, dx \] Input:
integrate(sec(d*x+c)**(1/2)/(-3-2*sec(d*x+c))**(1/2),x)
Output:
Integral(sqrt(sec(c + d*x))/sqrt(-2*sec(c + d*x) - 3), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {-2 \, \sec \left (d x + c\right ) - 3}} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)/(-3-2*sec(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(sec(d*x + c))/sqrt(-2*sec(d*x + c) - 3), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {-2 \, \sec \left (d x + c\right ) - 3}} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)/(-3-2*sec(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(sec(d*x + c))/sqrt(-2*sec(d*x + c) - 3), x)
Timed out. \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=\int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {-\frac {2}{\cos \left (c+d\,x\right )}-3}} \,d x \] Input:
int((1/cos(c + d*x))^(1/2)/(- 2/cos(c + d*x) - 3)^(1/2),x)
Output:
int((1/cos(c + d*x))^(1/2)/(- 2/cos(c + d*x) - 3)^(1/2), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx=-\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {-2 \sec \left (d x +c \right )-3}}{2 \sec \left (d x +c \right )+3}d x \right ) \] Input:
int(sec(d*x+c)^(1/2)/(-3-2*sec(d*x+c))^(1/2),x)
Output:
- int((sqrt(sec(c + d*x))*sqrt( - 2*sec(c + d*x) - 3))/(2*sec(c + d*x) + 3),x)