3.131 \(\int \sqrt{a+b \text{sech}(c+d x)} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 \coth (c+d x) \sqrt{-\frac{b (1-\text{sech}(c+d x))}{a+b \text{sech}(c+d x)}} \sqrt{\frac{b (\text{sech}(c+d x)+1)}{a+b \text{sech}(c+d x)}} (a+b \text{sech}(c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \text{sech}(c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]

[Out]

(2*Coth[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sech[c + d*x]]], (a - b)/(a + b)]*Sqrt[-(
(b*(1 - Sech[c + d*x]))/(a + b*Sech[c + d*x]))]*Sqrt[(b*(1 + Sech[c + d*x]))/(a + b*Sech[c + d*x])]*(a + b*Sec
h[c + d*x]))/(Sqrt[a + b]*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0259978, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3780} \[ \frac{2 \coth (c+d x) \sqrt{-\frac{b (1-\text{sech}(c+d x))}{a+b \text{sech}(c+d x)}} \sqrt{\frac{b (\text{sech}(c+d x)+1)}{a+b \text{sech}(c+d x)}} (a+b \text{sech}(c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \text{sech}(c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(2*Coth[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sech[c + d*x]]], (a - b)/(a + b)]*Sqrt[-(
(b*(1 - Sech[c + d*x]))/(a + b*Sech[c + d*x]))]*Sqrt[(b*(1 + Sech[c + d*x]))/(a + b*Sech[c + d*x])]*(a + b*Sec
h[c + d*x]))/(Sqrt[a + b]*d)

Rule 3780

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*(a + b*Csc[c + d*x])*Sqrt[(b*(1 + Csc[c +
d*x]))/(a + b*Csc[c + d*x])]*Sqrt[-((b*(1 - Csc[c + d*x]))/(a + b*Csc[c + d*x]))]*EllipticPi[a/(a + b), ArcSi
n[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)])/(d*Rt[a + b, 2]*Cot[c + d*x]), x] /; FreeQ[{a, b,
c, d}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \sqrt{a+b \text{sech}(c+d x)} \, dx &=\frac{2 \coth (c+d x) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \text{sech}(c+d x)}}\right )|\frac{a-b}{a+b}\right ) \sqrt{-\frac{b (1-\text{sech}(c+d x))}{a+b \text{sech}(c+d x)}} \sqrt{\frac{b (1+\text{sech}(c+d x))}{a+b \text{sech}(c+d x)}} (a+b \text{sech}(c+d x))}{\sqrt{a+b} d}\\ \end{align*}

Mathematica [F]  time = 7.71742, size = 0, normalized size = 0. \[ \int \sqrt{a+b \text{sech}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sech[c + d*x]],x]

[Out]

Integrate[Sqrt[a + b*Sech[c + d*x]], x]

________________________________________________________________________________________

Maple [F]  time = 0.194, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b{\rm sech} \left (dx+c\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sech(d*x + c) + a), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sech(d*x + c) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sech(c + d*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sech(d*x + c) + a), x)