Integrand size = 12, antiderivative size = 38 \[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \] Output:
2*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(b*sec(f*x+e))^ (1/2)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \] Input:
Integrate[1/Sqrt[b*Sec[e + f*x]],x]
Output:
(2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])
Time = 0.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4258, 3042, 3119}
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.
\(\displaystyle \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\) |
Input:
Int[1/Sqrt[b*Sec[e + f*x]],x]
Output:
(2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 4.61
method | result | size |
default | \(\frac {2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (2+\cos \left (f x +e \right )+\sec \left (f x +e \right )\right ) \operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )+2 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (-2-\cos \left (f x +e \right )-\sec \left (f x +e \right )\right )+2 \sin \left (f x +e \right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {b \sec \left (f x +e \right )}}\) | \(175\) |
risch | \(-\frac {i \sqrt {2}}{f \sqrt {\frac {b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {i \left (-\frac {2 \left (b \,{\mathrm e}^{2 i \left (f x +e \right )}+b \right )}{b \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (b \,{\mathrm e}^{2 i \left (f x +e \right )}+b \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 i \left (f x +e \right )} b +b \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {b \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}}{f \sqrt {\frac {b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(299\) |
Input:
int(1/(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/f/(cos(f*x+e)+1)/(b*sec(f*x+e))^(1/2)*(I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f *x+e)/(cos(f*x+e)+1))^(1/2)*(2+cos(f*x+e)+sec(f*x+e))*EllipticF(I*(cot(f*x +e)-csc(f*x+e)),I)+I*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(cos(f*x+e)+1))^ (1/2)*EllipticE(I*(cot(f*x+e)-csc(f*x+e)),I)*(-2-cos(f*x+e)-sec(f*x+e))+si n(f*x+e))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\frac {i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{b f} \] Input:
integrate(1/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
Output:
(I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f *x + e) + I*sin(f*x + e))) - I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, weie rstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(b*f)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {1}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \] Input:
integrate(1/(b*sec(f*x+e))**(1/2),x)
Output:
Integral(1/sqrt(b*sec(e + f*x)), x)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(b*sec(f*x + e)), x)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \] Input:
integrate(1/(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(b*sec(f*x + e)), x)
Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {1}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \] Input:
int(1/(b/cos(e + f*x))^(1/2),x)
Output:
int(1/(b/cos(e + f*x))^(1/2), x)
\[ \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sec \left (f x +e \right )}}{\sec \left (f x +e \right )}d x \right )}{b} \] Input:
int(1/(b*sec(f*x+e))^(1/2),x)
Output:
(sqrt(b)*int(sqrt(sec(e + f*x))/sec(e + f*x),x))/b