### 3.999 $$\int \frac{\text{sech}^2(x)}{\sqrt{1-4 \tanh ^2(x)}} \, dx$$

Optimal. Leaf size=9 $\frac{1}{2} \sin ^{-1}(2 \tanh (x))$

[Out]

ArcSin[2*Tanh[x]]/2

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Rubi [A]  time = 0.0495234, 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 = {3675, 216} $\frac{1}{2} \sin ^{-1}(2 \tanh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^2/Sqrt[1 - 4*Tanh[x]^2],x]

[Out]

ArcSin[2*Tanh[x]]/2

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 216

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

Rubi steps

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

Mathematica [B]  time = 0.053116, size = 47, normalized size = 5.22 $\frac{\sqrt{3 \cosh (2 x)-5} \text{sech}(x) \tanh ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3 \sinh ^2(x)-1}}\right )}{2 \sqrt{2-8 \tanh ^2(x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[x]^2/Sqrt[1 - 4*Tanh[x]^2],x]

[Out]

(ArcTanh[(2*Sinh[x])/Sqrt[-1 + 3*Sinh[x]^2]]*Sqrt[-5 + 3*Cosh[2*x]]*Sech[x])/(2*Sqrt[2 - 8*Tanh[x]^2])

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Maple [F]  time = 0.17, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm sech} \left (x\right ) \right ) ^{2}{\frac{1}{\sqrt{1-4\, \left ( \tanh \left ( x \right ) \right ) ^{2}}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (x\right )^{2}}{\sqrt{-4 \, \tanh \left (x\right )^{2} + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sech(x)^2/sqrt(-4*tanh(x)^2 + 1), x)

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Fricas [B]  time = 2.55978, size = 398, normalized size = 44.22 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{2 \, \sqrt{2}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt{-\frac{3 \, \cosh \left (x\right )^{2} + 3 \, \sinh \left (x\right )^{2} - 5}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{3 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 2 \,{\left (9 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \,{\left (3 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-(3*cosh(x)^2 + 3*sinh(x)^2 - 5)/(c
osh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(3*cosh(x)^4 + 12*cosh(x)*sinh(x)^3 + 3*sinh(x)^4 + 2*(9*cosh(x)^2
- 5)*sinh(x)^2 - 10*cosh(x)^2 + 4*(3*cosh(x)^3 - 5*cosh(x))*sinh(x) + 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{\sqrt{- \left (2 \tanh{\left (x \right )} - 1\right ) \left (2 \tanh{\left (x \right )} + 1\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(x)**2/sqrt(-(2*tanh(x) - 1)*(2*tanh(x) + 1)), x)

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Giac [B]  time = 1.21861, size = 59, normalized size = 6.56 \begin{align*} -\arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (\sqrt{3} \sqrt{-3 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 3} - 4\right )}}{3 \, e^{\left (2 \, x\right )} - 5} - 1\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/3*sqrt(3)*(2*(sqrt(3)*sqrt(-3*e^(4*x) + 10*e^(2*x) - 3) - 4)/(3*e^(2*x) - 5) - 1))