### 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]

(Cosh[x] + Sinh[x])^2/2

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Rubi [A]  time = 0.0389334, 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]

(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*}

Mathematica [A]  time = 0.0034965, 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., 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 = 1.02184, 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 [A]  time = 2.2091, size = 61, normalized size = 5.55 \begin{align*} \frac{\cosh \left (x\right ) + \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\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) + sinh(x))/(cosh(x) - sinh(x))

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Sympy [A]  time = 0.376119, size = 8, normalized size = 0.73 \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 = 1.14802, 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)