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*arctanh(tanh(x)/(1+sech(x))^(1/2))*3^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 19, 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 = {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 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])

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]

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*}

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Mathematica [B]  time = 0.04, 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]]

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fricas [B]  time = 0.38, size = 233, normalized size = 12.26 \[ \frac {1}{2} \, \sqrt {3} \log \left (-\frac {\cosh \relax (x)^{4} + {\left (4 \, \cosh \relax (x) - 3\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{3} + {\left (6 \, \cosh \relax (x)^{2} - 9 \, \cosh \relax (x) + 5\right )} \sinh \relax (x)^{2} + \sqrt {2} {\left (\cosh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x)^{2} + \sinh \relax (x)^{3} - 3 \, \cosh \relax (x)^{2} + {\left (3 \, \cosh \relax (x)^{2} - 6 \, \cosh \relax (x) + 4\right )} \sinh \relax (x) + 4 \, \cosh \relax (x) - 4\right )} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} + 5 \, \cosh \relax (x)^{2} + {\left (4 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)^{2} + 10 \, \cosh \relax (x) - 4\right )} \sinh \relax (x) - 4 \, \cosh \relax (x) + 4}{\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3}}\right ) + \frac {1}{2} \, \sqrt {3} \log \left (\frac {\sqrt {2} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x) + 1\right )} + \cosh \relax (x)^{2} + {\left (2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + \cosh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x)}\right ) \]

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)))

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giac [B]  time = 0.14, size = 52, normalized size = 2.74 \[ -\sqrt {3} {\left (\log \left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x} + 1\right ) + \log \left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right ) - \log \left (-\sqrt {e^{\left (2 \, x\right )} + 1} + e^{x} + 1\right )\right )} \]

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))

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maple [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \sqrt {3+3 \,\mathrm {sech}\relax (x )}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 \, \operatorname {sech}\relax (x) + 3}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \sqrt {\frac {3}{\mathrm {cosh}\relax (x)}+3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3/cosh(x) + 3)^(1/2),x)

[Out]

int((3/cosh(x) + 3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {3} \int \sqrt {\operatorname {sech}{\relax (x )} + 1}\, dx \]

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)

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