3.11 \(\int \sqrt {\text {sech}(a+b x)} \, dx\)

Optimal. Leaf size=40 \[ -\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \]

[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

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3771, 2641} \[ -\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[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 2641

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

Rule 3771

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*} \int \sqrt {\text {sech}(a+b x)} \, dx &=\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)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 40, normalized size = 1.00 \[ -\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[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

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\operatorname {sech}\left (b x + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(sech(b*x + a)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {sech}\left (b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sech(b*x + a)), x)

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maple [B]  time = 0.40, size = 135, normalized size = 3.38 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((2*cosh(1/2*b*x+1/2*a)^2-1)*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))/s
inh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {sech}\left (b x + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sech(b*x + a)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(a + b*x))^(1/2),x)

[Out]

int((1/cosh(a + b*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {sech}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sech(a + b*x)), x)

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