\(\int \sqrt {b \sec (e+f x)} \, dx\) [381]

Optimal result

Integrand size = 12, antiderivative size = 38 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \]

[Out]

2*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(b*se
c(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2720} \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \]

[In]

Int[Sqrt[b*Sec[e + f*x]],x]

[Out]

(2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/f

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx \\ & = \frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \]

[In]

Integrate[Sqrt[b*Sec[e + f*x]],x]

[Out]

(2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/f

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.03

method	result	size
default	\(\frac {2 i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sqrt {b \sec \left (f x +e \right )}}{f}\)	\(77\)

[In]

int((b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*I/f*(cos(f*x+e)+1)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(cot(f*x+e)-csc(f*
x+e)),I)*(b*sec(f*x+e))^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.50 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {-i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{f} \]

[In]

integrate((b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

(-I*sqrt(2)*sqrt(b)*weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(2)*sqrt(b)*weierstrassP
Inverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)))/f

Sympy [F]

\[ \int \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}}\, dx \]

[In]

integrate((b*sec(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(b*sec(e + f*x)), x)

Maxima [F]

\[ \int \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \,d x } \]

[In]

integrate((b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sec(f*x + e)), x)

Giac [F]

\[ \int \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \,d x } \]

[In]

integrate((b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(f*x + e)), x)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2\,\sqrt {\cos \left (e+f\,x\right )}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}\,\mathrm {F}\left (\frac {e}{2}+\frac {f\,x}{2}\middle |2\right )}{f} \]

[In]

int((b/cos(e + f*x))^(1/2),x)

[Out]

(2*cos(e + f*x)^(1/2)*(b/cos(e + f*x))^(1/2)*ellipticF(e/2 + (f*x)/2, 2))/f