3.34 \(\int \sqrt{a \text{sech}^2(x)} \, dx\)

Optimal. Leaf size=25 \[ \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right ) \]

[Out]

Sqrt[a]*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]]

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Rubi [A]  time = 0.0157782, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 217, 203} \[ \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Sech[x]^2],x]

[Out]

Sqrt[a]*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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{a \text{sech}^2(x)} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x^2}} \, dx,x,\tanh (x)\right )\\ &=a \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right )\\ &=\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a \text{sech}^2(x)}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0059364, size = 21, normalized size = 0.84 \[ 2 \cosh (x) \sqrt{a \text{sech}^2(x)} \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Sech[x]^2],x]

[Out]

2*ArcTan[Tanh[x/2]]*Cosh[x]*Sqrt[a*Sech[x]^2]

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Maple [C]  time = 0.082, size = 72, normalized size = 2.9 \begin{align*} i\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ) \ln \left ({{\rm e}^{x}}+i \right ) -i\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ) \ln \left ({{\rm e}^{x}}-i \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^2)^(1/2),x)

[Out]

I*(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*exp(-x)*(exp(2*x)+1)*ln(exp(x)+I)-I*(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*exp(
-x)*(exp(2*x)+1)*ln(exp(x)-I)

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Maxima [A]  time = 1.77084, size = 11, normalized size = 0.44 \begin{align*} 2 \, \sqrt{a} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)*arctan(e^x)

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Fricas [A]  time = 2.23395, size = 444, normalized size = 17.76 \begin{align*} \left [\sqrt{-a} \log \left (\frac{2 \, a \cosh \left (x\right ) e^{x} \sinh \left (x\right ) + a e^{x} \sinh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} +{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )\right )} \sqrt{-a} \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} +{\left (a \cosh \left (x\right )^{2} - a\right )} e^{x}}{2 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right ) + e^{x} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{x}}\right ), 2 \, \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (2 \, x\right )} + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(-a)*log((2*a*cosh(x)*e^x*sinh(x) + a*e^x*sinh(x)^2 + 2*(cosh(x)*e^(2*x) + (e^(2*x) + 1)*sinh(x) + cosh(x
))*sqrt(-a)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x + (a*cosh(x)^2 - a)*e^x)/(2*cosh(x)*e^x*sinh(x) + e^x*sinh(x
)^2 + (cosh(x)^2 + 1)*e^x)), 2*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1)*arctan(cosh(x) + sinh(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \operatorname{sech}^{2}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**2)**(1/2),x)

[Out]

Integral(sqrt(a*sech(x)**2), x)

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Giac [A]  time = 1.13218, size = 11, normalized size = 0.44 \begin{align*} 2 \, \sqrt{a} \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)*arctan(e^x)