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 = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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]

-1/2*Cosh[2*x] + Sinh[2*x]/2

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fricas [B]  time = 0.43, size = 19, normalized size = 1.73 \[ -\frac {1}{2 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]

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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giac [A]  time = 0.11, size = 6, normalized size = 0.55 \[ -\frac {1}{2} \, e^{\left (-2 \, x\right )} \]

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)

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maple [A]  time = 0.02, size = 17, normalized size = 1.55 \[ -\frac {\cosh \relax (x )-\sinh \relax (x )}{2 \left (\cosh \relax (x )+\sinh \relax (x )\right )} \]

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 = 0.40, size = 6, normalized size = 0.55 \[ -\frac {1}{2} \, e^{\left (-2 \, x\right )} \]

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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mupad [B]  time = 1.53, size = 6, normalized size = 0.55 \[ -\frac {{\mathrm {e}}^{-2\,x}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(-2*x)/2

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sympy [A]  time = 0.31, size = 10, normalized size = 0.91 \[ - \frac {\cosh {\relax (x )}}{\sinh {\relax (x )} + \cosh {\relax (x )}} \]

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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