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

Optimal. Leaf size=1 \[ x \]

[Out]

x

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Rubi [A]  time = 0.01, antiderivative size = 1, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4381, 8} \[ x \]

Antiderivative was successfully verified.

[In]

Int[(Sech[x]^2 + Tanh[x]^2)^(-2),x]

[Out]

x

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4381

Int[(u_.)*((a_.) + (c_.)*sec[(d_.) + (e_.)*(x_)]^2 + (b_.)*tan[(d_.) + (e_.)*(x_)]^2)^(p_.), x_Symbol] :> Dist
[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 1, normalized size = 1.00 \[ x \]

Antiderivative was successfully verified.

[In]

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

[Out]

x

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fricas [A]  time = 0.38, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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giac [A]  time = 0.11, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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maple [C]  time = 0.13, size = 8, normalized size = 8.00 \[ 2 \arctanh \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctanh(tanh(1/2*x))

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maxima [A]  time = 0.31, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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mupad [B]  time = 1.57, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/cosh(x)^2 + tanh(x)^2)^2,x)

[Out]

x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((tanh(x)**2 + sech(x)**2)**(-2), x)

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