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 = 0.0322492, antiderivative size = 3, normalized size of antiderivative = 1., 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 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 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])

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*}

Mathematica [A]  time = 0.0033085, size = 3, normalized size = 1. \[ \tan ^{-1}(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Tanh[x]]

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Maple [B]  time = 0.038, size = 116, normalized size = 38.7 \begin{align*} -2\,{\frac{\sqrt{2}}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+2\,{\frac{\sqrt{2}}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) } \end{align*}

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 = 1.51455, size = 47, normalized size = 15.67 \begin{align*} \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 ) \end{align*}

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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Fricas [B]  time = 1.9952, size = 69, normalized size = 23. \begin{align*} -\arctan \left (-\frac{\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end{align*}

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

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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Giac [A]  time = 1.13488, size = 7, normalized size = 2.33 \begin{align*} \arctan \left (e^{\left (2 \, x\right )}\right ) \end{align*}

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))