\(\int \sqrt {\text {sech}(a+b x)} \, dx\) [11]

Optimal result

Integrand size = 10, antiderivative size = 40 \[ \int \sqrt {\text {sech}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b} \]

[Out]

-2*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh(1/2*a+1/2*b*x)*EllipticF(I*sinh(1/2*a+1/2*b*x),2^(1/2))*cosh(b*x+a)^(1
/2)*sech(b*x+a)^(1/2)/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3856, 2720} \[ \int \sqrt {\text {sech}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{b} \]

[In]

Int[Sqrt[Sech[a + b*x]],x]

[Out]

((-2*I)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b

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 {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx \\ & = -\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \sqrt {\text {sech}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{b} \]

[In]

Integrate[Sqrt[Sech[a + b*x]],x]

[Out]

((-2*I)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(62)=124\).

Time = 0.47 (sec) , antiderivative size = 135, normalized size of antiderivative = 3.38

method	result	size
default	\(\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\)	\(135\)

[In]

int(sech(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*((-1+2*cosh(1/2*b*x+1/2*a)^2)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1
/2*a)^2+1)^(1/2)/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))/
sinh(1/2*b*x+1/2*a)/(-1+2*cosh(1/2*b*x+1/2*a)^2)^(1/2)/b

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.60 \[ \int \sqrt {\text {sech}(a+b x)} \, dx=\frac {2 \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \]

[In]

integrate(sech(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(2)*weierstrassPInverse(-4, 0, cosh(b*x + a) + sinh(b*x + a))/b

Sympy [F]

\[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int \sqrt {\operatorname {sech}{\left (a + b x \right )}}\, dx \]

[In]

integrate(sech(b*x+a)**(1/2),x)

[Out]

Integral(sqrt(sech(a + b*x)), x)

Maxima [F]

\[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int { \sqrt {\operatorname {sech}\left (b x + a\right )} \,d x } \]

[In]

integrate(sech(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sech(b*x + a)), x)

Giac [F]

\[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int { \sqrt {\operatorname {sech}\left (b x + a\right )} \,d x } \]

[In]

integrate(sech(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sech(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\text {sech}(a+b x)} \, dx=\int \sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]

[In]

int((1/cosh(a + b*x))^(1/2),x)

[Out]

int((1/cosh(a + b*x))^(1/2), x)