### 3.998 $$\int \frac{\text{sech}^2(x)}{\sqrt{4-\text{sech}^2(x)}} \, dx$$

Optimal. Leaf size=9 $\sinh ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )$

[Out]

ArcSinh[Tanh[x]/Sqrt[3]]

________________________________________________________________________________________

Rubi [A]  time = 0.0494739, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {4146, 215} $\sinh ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^2/Sqrt[4 - Sech[x]^2],x]

[Out]

ArcSinh[Tanh[x]/Sqrt[3]]

Rule 4146

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]

Rule 215

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

Rubi steps

\begin{align*} \int \frac{\text{sech}^2(x)}{\sqrt{4-\text{sech}^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3+x^2}} \, dx,x,\tanh (x)\right )\\ &=\sinh ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [B]  time = 0.0462612, size = 43, normalized size = 4.78 $\frac{\sqrt{2 \cosh (2 x)+1} \text{sech}(x) \tanh ^{-1}\left (\frac{\sinh (x)}{\sqrt{4 \sinh ^2(x)+3}}\right )}{\sqrt{4-\text{sech}^2(x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[x]^2/Sqrt[4 - Sech[x]^2],x]

[Out]

(ArcTanh[Sinh[x]/Sqrt[3 + 4*Sinh[x]^2]]*Sqrt[1 + 2*Cosh[2*x]]*Sech[x])/Sqrt[4 - Sech[x]^2]

________________________________________________________________________________________

Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm sech} \left (x\right ) \right ) ^{2}{\frac{1}{\sqrt{4- \left ({\rm sech} \left (x\right ) \right ) ^{2}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2/(4-sech(x)^2)^(1/2),x)

[Out]

int(sech(x)^2/(4-sech(x)^2)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (x\right )^{2}}{\sqrt{-\operatorname{sech}\left (x\right )^{2} + 4}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(4-sech(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sech(x)^2/sqrt(-sech(x)^2 + 4), x)

________________________________________________________________________________________

Fricas [B]  time = 2.40214, size = 367, normalized size = 40.78 \begin{align*} -\log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt{\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}\right ) + \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt{\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(4-sech(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-log(-cosh(x)^2 - 2*cosh(x)*sinh(x) - sinh(x)^2 + sqrt((2*cosh(x)^2 + 2*sinh(x)^2 + 1)/(cosh(x)^2 - 2*cosh(x)*
sinh(x) + sinh(x)^2))) + log(-cosh(x)^2 - 2*cosh(x)*sinh(x) - sinh(x)^2 + sqrt((2*cosh(x)^2 + 2*sinh(x)^2 + 1)
/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 2)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(4-sech(x)**2)**(1/2),x)

[Out]

Integral(sech(x)**2/sqrt(-(sech(x) - 2)*(sech(x) + 2)), x)

________________________________________________________________________________________

Giac [B]  time = 1.16656, size = 59, normalized size = 6.56 \begin{align*} -\log \left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + \log \left (-\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(4-sech(x)^2)^(1/2),x, algorithm="giac")

[Out]

-log(sqrt(e^(4*x) + e^(2*x) + 1) - e^(2*x)) + log(-sqrt(e^(4*x) + e^(2*x) + 1) + e^(2*x) + 2)