### 3.983 $$\int \text{sech}^2(x) (1+\frac{1}{1-\tanh ^2(x)}) \, dx$$

Optimal. Leaf size=4 $x+\tanh (x)$

[Out]

x + Tanh[x]

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Rubi [A]  time = 0.0523521, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {206} $x+\tanh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^2*(1 + (1 - Tanh[x]^2)^(-1)),x]

[Out]

x + Tanh[x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0063401, size = 4, normalized size = 1. $x+\tanh (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x + Tanh[x]

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Maple [B]  time = 0.032, size = 34, normalized size = 8.5 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(tanh(1/2*x)+1)-ln(tanh(1/2*x)-1)+2*tanh(1/2*x)/(tanh(1/2*x)^2+1)

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Maxima [B]  time = 1.02392, size = 16, normalized size = 4. \begin{align*} x + \frac{2}{e^{\left (-2 \, x\right )} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2/(e^(-2*x) + 1)

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Fricas [B]  time = 2.31436, size = 50, normalized size = 12.5 \begin{align*} \frac{{\left (x - 1\right )} \cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x - 1)*cosh(x) + sinh(x))/cosh(x)

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Sympy [B]  time = 1.48378, size = 29, normalized size = 7.25 \begin{align*} - \frac{x \operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} - 1} - \frac{\tanh{\left (x \right )} \operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15623, size = 16, normalized size = 4. \begin{align*} x - \frac{2}{e^{\left (2 \, x\right )} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2/(e^(2*x) + 1)