Integrand size = 25, antiderivative size = 61 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=\frac {2 \sqrt {3+2 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-2-3 \sec (c+d x)}} \] Output:
2/5*(3+2*cos(d*x+c))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2/5*5^(1/2))*sec( d*x+c)^(1/2)*5^(1/2)/d/(-2-3*sec(d*x+c))^(1/2)
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=\frac {2 \sqrt {3+2 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-2-3 \sec (c+d x)}} \] Input:
Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-2 - 3*Sec[c + d*x]],x]
Output:
(2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]] )/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4345, 3042, 3142, 3042, 3140}
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.
\(\displaystyle \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3 \sec (c+d x)-2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \csc \left (c+d x+\frac {\pi }{2}\right )-2}}dx\) |
\(\Big \downarrow \) 4345 |
\(\displaystyle \frac {\sqrt {-2 \cos (c+d x)-3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-2 \cos (c+d x)-3}}dx}{\sqrt {-3 \sec (c+d x)-2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-2 \cos (c+d x)-3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {-2 \sin \left (c+d x+\frac {\pi }{2}\right )-3}}dx}{\sqrt {-3 \sec (c+d x)-2}}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {\sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\frac {2}{5} \cos (c+d x)+\frac {3}{5}}}dx}{\sqrt {5} \sqrt {-3 \sec (c+d x)-2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\frac {2}{5} \sin \left (c+d x+\frac {\pi }{2}\right )+\frac {3}{5}}}dx}{\sqrt {5} \sqrt {-3 \sec (c+d x)-2}}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {4}{5}\right )}{\sqrt {5} d \sqrt {-3 \sec (c+d x)-2}}\) |
Input:
Int[Sqrt[Sec[c + d*x]]/Sqrt[-2 - 3*Sec[c + d*x]],x]
Output:
(2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]] )/(Sqrt[5]*d*Sqrt[-2 - 3*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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], 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 = 119, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {i \sqrt {-2-3 \sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \frac {\sqrt {5}}{5}\right ) \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right )}{5 d \left (3+2 \cos \left (d x +c \right )\right )}\) | \(119\) |
Input:
int(sec(d*x+c)^(1/2)/(-2-3*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5*I/d*(-2-3*sec(d*x+c))^(1/2)*sec(d*x+c)^(1/2)*2^(1/2)*(1/(1+cos(d*x+c)) )^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(I*(csc( d*x+c)-cot(d*x+c)),1/5*5^(1/2))/(3+2*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 = 44, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=-\frac {{\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) + {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right )}{d} \] Input:
integrate(sec(d*x+c)^(1/2)/(-2-3*sec(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-(weierstrassPInverse(8, -4, cos(d*x + c) + I*sin(d*x + c) + 1) + weierstr assPInverse(8, -4, cos(d*x + c) - I*sin(d*x + c) + 1))/d
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=\int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {- 3 \sec {\left (c + d x \right )} - 2}}\, dx \] Input:
integrate(sec(d*x+c)**(1/2)/(-2-3*sec(d*x+c))**(1/2),x)
Output:
Integral(sqrt(sec(c + d*x))/sqrt(-3*sec(c + d*x) - 2), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {-3 \, \sec \left (d x + c\right ) - 2}} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)/(-2-3*sec(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(sec(d*x + c))/sqrt(-3*sec(d*x + c) - 2), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {-3 \, \sec \left (d x + c\right ) - 2}} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)/(-2-3*sec(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(sec(d*x + c))/sqrt(-3*sec(d*x + c) - 2), x)
Timed out. \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=\int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {-\frac {3}{\cos \left (c+d\,x\right )}-2}} \,d x \] Input:
int((1/cos(c + d*x))^(1/2)/(- 3/cos(c + d*x) - 2)^(1/2),x)
Output:
int((1/cos(c + d*x))^(1/2)/(- 3/cos(c + d*x) - 2)^(1/2), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx=-\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {-3 \sec \left (d x +c \right )-2}}{3 \sec \left (d x +c \right )+2}d x \right ) \] Input:
int(sec(d*x+c)^(1/2)/(-2-3*sec(d*x+c))^(1/2),x)
Output:
- int((sqrt(sec(c + d*x))*sqrt( - 3*sec(c + d*x) - 2))/(3*sec(c + d*x) + 2),x)