### 3.731 $$\int \frac{\cosh (x)-\sinh (x)}{\cosh (x)+\sinh (x)} \, dx$$

Optimal. Leaf size=11 $-\frac{1}{2 (\sinh (x)+\cosh (x))^2}$

[Out]

-1/(2*(Cosh[x] + Sinh[x])^2)

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Rubi [A]  time = 0.0344315, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {4385} $-\frac{1}{2 (\sinh (x)+\cosh (x))^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]),x]

[Out]

-1/(2*(Cosh[x] + Sinh[x])^2)

Rule 4385

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, Simp[(q*A
ctivateTrig[y^(m + 1)])/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1] &&  !InertTrigFreeQ[u]

Rubi steps

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

Mathematica [A]  time = 0.0049873, size = 17, normalized size = 1.55 $\frac{1}{2} \sinh (2 x)-\frac{1}{2} \cosh (2 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]),x]

[Out]

-Cosh[2*x]/2 + Sinh[2*x]/2

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Maple [A]  time = 0.002, size = 17, normalized size = 1.6 \begin{align*} -{\frac{\cosh \left ( x \right ) -\sinh \left ( x \right ) }{2\,\cosh \left ( x \right ) +2\,\sinh \left ( x \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)-sinh(x))/(cosh(x)+sinh(x)),x)

[Out]

-1/2*(cosh(x)-sinh(x))/(cosh(x)+sinh(x))

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Maxima [A]  time = 1.08188, size = 8, normalized size = 0.73 \begin{align*} -\frac{1}{2} \, e^{\left (-2 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-sinh(x))/(cosh(x)+sinh(x)),x, algorithm="maxima")

[Out]

-1/2*e^(-2*x)

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Fricas [B]  time = 2.28627, size = 68, normalized size = 6.18 \begin{align*} -\frac{1}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-sinh(x))/(cosh(x)+sinh(x)),x, algorithm="fricas")

[Out]

-1/2/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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Sympy [A]  time = 0.331629, size = 10, normalized size = 0.91 \begin{align*} - \frac{\cosh{\left (x \right )}}{\sinh{\left (x \right )} + \cosh{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-sinh(x))/(cosh(x)+sinh(x)),x)

[Out]

-cosh(x)/(sinh(x) + cosh(x))

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Giac [A]  time = 1.12036, size = 8, normalized size = 0.73 \begin{align*} -\frac{1}{2} \, e^{\left (-2 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((cosh(x)-sinh(x))/(cosh(x)+sinh(x)),x, algorithm="giac")

[Out]

-1/2*e^(-2*x)