3.85 \(\int \sqrt{3+3 \text{sech}(x)} \, dx\)

Optimal. Leaf size=19 \[ 2 \sqrt{3} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{\text{sech}(x)+1}}\right ) \]

[Out]

2*Sqrt[3]*ArcTanh[Tanh[x]/Sqrt[1 + Sech[x]]]

________________________________________________________________________________________

Rubi [A]  time = 0.0173795, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3774, 203} \[ 2 \sqrt{3} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{\text{sech}(x)+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 3*Sech[x]],x]

[Out]

2*Sqrt[3]*ArcTanh[Tanh[x]/Sqrt[1 + Sech[x]]]

Rule 3774

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*C
ot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{3+3 \text{sech}(x)} \, dx &=6 i \operatorname{Subst}\left (\int \frac{1}{3+x^2} \, dx,x,-\frac{3 i \tanh (x)}{\sqrt{3+3 \text{sech}(x)}}\right )\\ &=2 \sqrt{3} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{1+\text{sech}(x)}}\right )\\ \end{align*}

Mathematica [B]  time = 0.0381295, size = 39, normalized size = 2.05 \[ \sqrt{6} \sinh ^{-1}\left (\sqrt{2} \sinh \left (\frac{x}{2}\right )\right ) \sqrt{\cosh (x)} \text{sech}\left (\frac{x}{2}\right ) \sqrt{\text{sech}(x)+1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 + 3*Sech[x]],x]

[Out]

Sqrt[6]*ArcSinh[Sqrt[2]*Sinh[x/2]]*Sqrt[Cosh[x]]*Sech[x/2]*Sqrt[1 + Sech[x]]

________________________________________________________________________________________

Maple [F]  time = 0.116, size = 0, normalized size = 0. \begin{align*} \int \sqrt{3+3\,{\rm sech} \left (x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+3*sech(x))^(1/2),x)

[Out]

int((3+3*sech(x))^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{3 \, \operatorname{sech}\left (x\right ) + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+3*sech(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(3*sech(x) + 3), x)

________________________________________________________________________________________

Fricas [B]  time = 2.4964, size = 833, normalized size = 43.84 \begin{align*} \frac{1}{2} \, \sqrt{3} \log \left (-\frac{\cosh \left (x\right )^{4} +{\left (4 \, \cosh \left (x\right ) - 3\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} +{\left (6 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{2} + \sqrt{2}{\left (\cosh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} - 6 \, \cosh \left (x\right ) + 4\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) - 4\right )} \sqrt{\frac{\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} + 5 \, \cosh \left (x\right )^{2} +{\left (4 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) - 4\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 4}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}}\right ) + \frac{1}{2} \, \sqrt{3} \log \left (\frac{\sqrt{2} \sqrt{\frac{\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )} + \cosh \left (x\right )^{2} +{\left (2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \cosh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+3*sech(x))^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(3)*log(-(cosh(x)^4 + (4*cosh(x) - 3)*sinh(x)^3 + sinh(x)^4 - 3*cosh(x)^3 + (6*cosh(x)^2 - 9*cosh(x) +
5)*sinh(x)^2 + sqrt(2)*(cosh(x)^3 + 3*(cosh(x) - 1)*sinh(x)^2 + sinh(x)^3 - 3*cosh(x)^2 + (3*cosh(x)^2 - 6*co
sh(x) + 4)*sinh(x) + 4*cosh(x) - 4)*sqrt(cosh(x)/(cosh(x) - sinh(x))) + 5*cosh(x)^2 + (4*cosh(x)^3 - 9*cosh(x)
^2 + 10*cosh(x) - 4)*sinh(x) - 4*cosh(x) + 4)/(cosh(x)^3 + 3*cosh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)
^3)) + 1/2*sqrt(3)*log((sqrt(2)*sqrt(cosh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x) + 1) + cosh(x)^2 + (2*cos
h(x) + 1)*sinh(x) + sinh(x)^2 + cosh(x) + 1)/(cosh(x) + sinh(x)))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{3} \int \sqrt{\operatorname{sech}{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+3*sech(x))**(1/2),x)

[Out]

sqrt(3)*Integral(sqrt(sech(x) + 1), x)

________________________________________________________________________________________

Giac [B]  time = 1.14755, size = 70, normalized size = 3.68 \begin{align*} -\sqrt{3}{\left (\log \left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x} + 1\right ) + \log \left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) - \log \left (-\sqrt{e^{\left (2 \, x\right )} + 1} + e^{x} + 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+3*sech(x))^(1/2),x, algorithm="giac")

[Out]

-sqrt(3)*(log(sqrt(e^(2*x) + 1) - e^x + 1) + log(sqrt(e^(2*x) + 1) - e^x) - log(-sqrt(e^(2*x) + 1) + e^x + 1))