3.1.18 \(\int \sqrt {b \text {sech}(c+d x)} \, dx\) [18]

Optimal. Leaf size=42 \[ -\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2720} \begin {gather*} -\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sech[c + d*x]],x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 42, normalized size = 1.00 \begin {gather*} -\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sech[c + d*x]],x]

[Out]

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

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Maple [F]
time = 1.54, size = 0, normalized size = 0.00 \[\int \sqrt {b \,\mathrm {sech}\left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sech(d*x + c)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 27, normalized size = 0.64 \begin {gather*} \frac {2 \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {b \operatorname {sech}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*sech(c + d*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sech(d*x + c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/cosh(c + d*x))^(1/2),x)

[Out]

int((b/cosh(c + d*x))^(1/2), x)

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