3.1.85 \(\int \sqrt {3+3 \text {sech}(x)} \, dx\) [85]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {3859, 209} \begin {gather*} 2 \sqrt {3} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {\text {sech}(x)+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 3*Sech[x]],x]

[Out]

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

Rule 209

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

Rule 3859

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 \text {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] Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(19)=38\).
time = 0.03, size = 39, normalized size = 2.05 \begin {gather*} \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 {1+\text {sech}(x)} \end {gather*}

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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Maple [F]
time = 1.21, size = 0, normalized size = 0.00 \[\int \sqrt {3+3 \,\mathrm {sech}\left (x \right )}\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (15) = 30\).
time = 0.40, size = 233, normalized size = 12.26 \begin {gather*} \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 {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \sqrt {3} \int \sqrt {\operatorname {sech}{\left (x \right )} + 1}\, dx \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (15) = 30\).
time = 0.39, size = 52, normalized size = 2.74 \begin {gather*} -\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 {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \sqrt {\frac {3}{\mathrm {cosh}\left (x\right )}+3} \,d x \end {gather*}

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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