### 3.816 $$\int \frac{1}{(\text{sech}^2(x)+\tanh ^2(x))^3} \, dx$$

Optimal. Leaf size=1 $x$

[Out]

x

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Rubi [A]  time = 0.0129531, antiderivative size = 1, normalized size of antiderivative = 1., 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)^(-3),x]

[Out]

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]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{\left (\text{sech}^2(x)+\tanh ^2(x)\right )^3} \, dx &=\int 1 \, dx\\ &=x\\ \end{align*}

Mathematica [A]  time = 0.0003534, size = 1, normalized size = 1. $x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Sech[x]^2 + Tanh[x]^2)^(-3),x]

[Out]

x

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Maple [C]  time = 0.024, size = 8, normalized size = 8. \begin{align*} 2\,{\it Artanh} \left ( \tanh \left ( x/2 \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sech(x)^2+tanh(x)^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.10592, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(x)^2+tanh(x)^2)^3,x, algorithm="maxima")

[Out]

x

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Fricas [A]  time = 2.2931, size = 4, normalized size = 4. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(x)^2+tanh(x)^2)^3,x, algorithm="fricas")

[Out]

x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\tanh ^{2}{\left (x \right )} + \operatorname{sech}^{2}{\left (x \right )}\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(x)**2+tanh(x)**2)**3,x)

[Out]

Integral((tanh(x)**2 + sech(x)**2)**(-3), x)

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Giac [A]  time = 1.14536, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(x)^2+tanh(x)^2)^3,x, algorithm="giac")

[Out]

x