3.9.11 \(\int \frac {1}{\cosh ^2(x)-\sinh ^2(x)} \, dx\) [811]

Optimal. Leaf size=1 \[ x \]

[Out]

x

________________________________________________________________________________________

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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4465, 8} \begin {gather*} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^2 - Sinh[x]^2)^(-1),x]

[Out]

x

Rule 8

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

Rule 4465

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]

Rubi steps

\begin {align*} \int \frac {1}{\cosh ^2(x)-\sinh ^2(x)} \, dx &=\int 1 \, dx\\ &=x\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^2 - Sinh[x]^2)^(-1),x]

[Out]

x

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.72, size = 8, normalized size = 8.00

method	result	size
risch	\(x\)	\(2\)
default	\(2 \arctanh \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2-sinh(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

________________________________________________________________________________________

Fricas [A]
time = 0.46, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (0) = 0\).
time = 0.19, size = 10, normalized size = 10.00 \begin {gather*} \frac {x}{- \sinh ^{2}{\left (x \right )} + \cosh ^{2}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(-sinh(x)**2 + cosh(x)**2)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

________________________________________________________________________________________

Mupad [B]
time = 1.52, size = 1, normalized size = 1.00 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2 - sinh(x)^2),x)

[Out]

x

________________________________________________________________________________________