3.980 \(\int \frac {\text {sech}^2(x)}{1+\tanh ^2(x)} \, dx\)

Optimal. Leaf size=3 \[ \tan ^{-1}(\tanh (x)) \]

[Out]

arctan(tanh(x))

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Rubi [A]  time = 0.03, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3675, 203} \[ \tan ^{-1}(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/(1 + Tanh[x]^2),x]

[Out]

ArcTan[Tanh[x]]

Rule 203

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

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])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Tanh[x]]

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fricas [B]  time = 0.44, size = 19, normalized size = 6.33 \[ -\arctan \left (-\frac {\cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(-(cosh(x) + sinh(x))/(cosh(x) - sinh(x)))

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giac [A]  time = 0.12, size = 5, normalized size = 1.67 \[ \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^(2*x))

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maple [B]  time = 0.23, size = 116, normalized size = 38.67 \[ \frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.40, size = 35, normalized size = 11.67 \[ \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^(-x))) - arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^(-x)))

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mupad [B]  time = 0.07, size = 5, normalized size = 1.67 \[ \mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2*(tanh(x)^2 + 1)),x)

[Out]

atan(exp(2*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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