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 = 0.05, antiderivative size = 4, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ x+\tanh (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + Tanh[x]

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fricas [B]  time = 0.42, size = 14, normalized size = 3.50 \[ \frac {{\left (x - 1\right )} \cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x)} \]

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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giac [B]  time = 0.13, size = 12, normalized size = 3.00 \[ x - \frac {2}{e^{\left (2 \, x\right )} + 1} \]

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)

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maple [B]  time = 0.17, size = 34, normalized size = 8.50 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {2 \tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1} \]

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 = 0.30, size = 12, normalized size = 3.00 \[ x + \frac {2}{e^{\left (-2 \, x\right )} + 1} \]

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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mupad [B]  time = 1.72, size = 12, normalized size = 3.00 \[ x-\frac {2}{{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2/(exp(2*x) + 1)

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sympy [B]  time = 0.81, size = 29, normalized size = 7.25 \[ - \frac {x \operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} - 1} - \frac {\tanh {\relax (x )} \operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} - 1} \]

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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