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

Optimal. Leaf size=20 \[ e^x x-e^x+\frac{e^{2 x}}{2} \]

[Out]

-E^x + E^(2*x)/2 + E^x*x

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Rubi [A]  time = 0.1087, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ e^x x-e^x+\frac{e^{2 x}}{2} \]

Antiderivative was successfully verified.

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

[Out]

-E^x + E^(2*x)/2 + E^x*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (x + e^{x}\right )^{2}}{2} - e^{x} - \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+cosh(x)+sinh(x))/(cosh(x)-sinh(x)),x)

[Out]

(x + exp(x))**2/2 - exp(x) - Integral(x, x)

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Mathematica [A]  time = 0.0641025, size = 23, normalized size = 1.15 \[ (x-1) \sinh (x)+\frac{1}{2} \cosh (2 x)+(x+\sinh (x)-1) \cosh (x) \]

Antiderivative was successfully verified.

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

[Out]

Cosh[2*x]/2 + (-1 + x)*Sinh[x] + Cosh[x]*(-1 + x + Sinh[x])

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Maple [A]  time = 0.112, size = 14, normalized size = 0.7 \[ \left ( -1+x \right ){{\rm e}^{x}}+{\frac{{{\rm e}^{2\,x}}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+cosh(x)+sinh(x))/(cosh(x)-sinh(x)),x)

[Out]

(-1+x)*exp(x)+1/2*exp(2*x)

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Maxima [A]  time = 1.3682, size = 18, normalized size = 0.9 \[{\left (x - 1\right )} e^{x} + \frac{1}{2} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)*e^x + 1/2*e^(2*x)

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Fricas [A]  time = 0.215303, size = 27, normalized size = 1.35 \[ \frac{2 \, x + \cosh \left (x\right ) + \sinh \left (x\right ) - 2}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.662627, size = 26, normalized size = 1.3 \[ \frac{x}{- \sinh{\left (x \right )} + \cosh{\left (x \right )}} + \frac{\sinh{\left (x \right )}}{- \sinh{\left (x \right )} + \cosh{\left (x \right )}} - \frac{1}{- \sinh{\left (x \right )} + \cosh{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x+cosh(x)+sinh(x))/(cosh(x)-sinh(x)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218583, size = 15, normalized size = 0.75 \[ \frac{1}{2} \,{\left (2 \, x + e^{x} - 2\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*x + e^x - 2)*e^x