### 3.813 $$\int \frac{1}{(\cosh ^2(x)-\sinh ^2(x))^3} \, dx$$

Optimal. Leaf size=1 $x$

[Out]

x

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Rubi [A]  time = 0.0135388, antiderivative size = 1, 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 = {4380, 8} $x$

Antiderivative was successfully veriﬁed.

[In]

Int[(Cosh[x]^2 - Sinh[x]^2)^(-3),x]

[Out]

x

Rule 4380

Int[(u_.)*((a_.) + cos[(d_.) + (e_.)*(x_)]^2*(b_.) + (c_.)*sin[(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 (\cosh ^2(x)-\sinh ^2(x)\right )^3} \, dx &=\int 1 \, dx\\ &=x\\ \end{align*}

Mathematica [A]  time = 0.0003531, size = 1, normalized size = 1. $x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Cosh[x]^2 - Sinh[x]^2)^(-3),x]

[Out]

x

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Maple [C]  time = 0.013, 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/(cosh(x)^2-sinh(x)^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.15499, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)^2-sinh(x)^2)^3,x, algorithm="maxima")

[Out]

x

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Fricas [A]  time = 2.22723, size = 4, normalized size = 4. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)^2-sinh(x)^2)^3,x, algorithm="fricas")

[Out]

x

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Sympy [B]  time = 4.73535, size = 34, normalized size = 34. \begin{align*} \frac{x}{- \sinh ^{6}{\left (x \right )} + 3 \sinh ^{4}{\left (x \right )} \cosh ^{2}{\left (x \right )} - 3 \sinh ^{2}{\left (x \right )} \cosh ^{4}{\left (x \right )} + \cosh ^{6}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)**2-sinh(x)**2)**3,x)

[Out]

x/(-sinh(x)**6 + 3*sinh(x)**4*cosh(x)**2 - 3*sinh(x)**2*cosh(x)**4 + cosh(x)**6)

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Giac [A]  time = 1.11921, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)^2-sinh(x)^2)^3,x, algorithm="giac")

[Out]

x