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]

arctan(a^(1/2)*tanh(x)/(a*sech(x)^2)^(1/2))*a^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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 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 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
]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.44, size = 145, normalized size = 5.80 \[ \left [\sqrt {-a} \log \left (\frac {2 \, a \cosh \relax (x) e^{x} \sinh \relax (x) + a e^{x} \sinh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + {\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x) + \cosh \relax (x)\right )} \sqrt {-a} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} + {\left (a \cosh \relax (x)^{2} - a\right )} e^{x}}{2 \, \cosh \relax (x) e^{x} \sinh \relax (x) + e^{x} \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{x}}\right ), 2 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right )\right ] \]

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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giac [A]  time = 0.13, size = 8, normalized size = 0.32 \[ 2 \, \sqrt {a} \arctan \left (e^{x}\right ) \]

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)

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maple [C]  time = 0.22, size = 72, normalized size = 2.88 \[ i \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{x}+i\right )-i \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{x}-i\right ) \]

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

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maxima [A]  time = 0.48, size = 8, normalized size = 0.32 \[ 2 \, \sqrt {a} \arctan \left (e^{x}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {\frac {a}{{\mathrm {cosh}\relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cosh(x)^2)^(1/2),x)

[Out]

int((a/cosh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}^{2}{\relax (x )}}\, dx \]

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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