3.730 \(\int \frac {\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx\)

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 = 0.04, 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]

(Cosh[x] + Sinh[x])^2/2

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]

Cosh[2*x]/2 + Sinh[2*x]/2

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fricas [A]  time = 0.53, size = 16, normalized size = 1.45 \[ \frac {\cosh \relax (x) + \sinh \relax (x)}{2 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 \[ \frac {\cosh \relax (x )+\sinh \relax (x )}{2 \cosh \relax (x )-2 \sinh \relax (x )} \]

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 = 0.50, 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 = 0.06, 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 = 0.35, size = 8, normalized size = 0.73 \[ \frac {\cosh {\relax (x )}}{- \sinh {\relax (x )} + \cosh {\relax (x )}} \]

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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