### 3.817 $$\int \frac{1}{\text{sech}^2(x)-\tanh ^2(x)} \, dx$$

Optimal. Leaf size=19 $\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x$

[Out]

-x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0282499, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {1093, 207} $\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x$

Antiderivative was successfully veriﬁed.

[In]

Int[(Sech[x]^2 - Tanh[x]^2)^(-1),x]

[Out]

-x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0894322, size = 19, normalized size = 1. $\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Sech[x]^2 - Tanh[x]^2)^(-1),x]

[Out]

-x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]

________________________________________________________________________________________

Maple [B]  time = 0.041, size = 54, normalized size = 2.8 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(tanh(1/2*x)+1)+2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)+2)*2^(1/2))+ln(tanh(1/2*x)-1)+2^(1/2)*arctanh(1/4*(2*tan
h(1/2*x)-2)*2^(1/2))

________________________________________________________________________________________

Maxima [B]  time = 1.67006, size = 86, normalized size = 4.53 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) - x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(-(sqrt(2) - e^(-x) + 1)/(sqrt(2) + e^(-x) - 1)) - 1/2*sqrt(2)*log(-(sqrt(2) - e^(-x) - 1)/(sqr
t(2) + e^(-x) + 1)) - x

________________________________________________________________________________________

Fricas [B]  time = 2.40961, size = 220, normalized size = 11.58 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{3 \,{\left (2 \, \sqrt{2} - 3\right )} \cosh \left (x\right )^{2} - 4 \,{\left (3 \, \sqrt{2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (2 \, \sqrt{2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(-(3*(2*sqrt(2) - 3)*cosh(x)^2 - 4*(3*sqrt(2) - 4)*cosh(x)*sinh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^
2 - 2*sqrt(2) + 3)/(cosh(x)^2 + sinh(x)^2 - 3)) - x

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.15631, size = 55, normalized size = 2.89 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\frac{{\left | -4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6)) - x