3.10.80 \(\int \frac {\text {sech}^2(x)}{1+\tanh ^2(x)} \, dx\) [980]

Optimal. Leaf size=3 \[ \text {ArcTan}(\tanh (x)) \]

[Out]

arctan(tanh(x))

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Rubi [A]
time = 0.02, 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 = {3756, 209} \begin {gather*} \text {ArcTan}(\tanh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/(1 + Tanh[x]^2),x]

[Out]

ArcTan[Tanh[x]]

Rule 209

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

Rule 3756

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 &=\text {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 = 0.01, size = 3, normalized size = 1.00 \begin {gather*} \text {ArcTan}(\tanh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Tanh[x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(3)=6\).
time = 1.19, size = 72, normalized size = 24.00

method	result	size
risch	\(\frac {i \ln \left ({\mathrm e}^{2 x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{2 x}-i\right )}{2}\)	\(24\)
default	\(-\frac {\left (2+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}-\frac {\left (-2+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (3) = 6\).
time = 0.49, size = 35, normalized size = 11.67 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (3) = 6\).
time = 0.34, size = 19, normalized size = 6.33 \begin {gather*} -\arctan \left (-\frac {\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} + 1}\, dx \end {gather*}

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 = 0.41, size = 5, normalized size = 1.67 \begin {gather*} \arctan \left (e^{\left (2 \, x\right )}\right ) \end {gather*}

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

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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