3.226 \(\int x \cosh (x) \, dx\)

Optimal. Leaf size=9 \[ x \sinh (x)-\cosh (x) \]

[Out]

-Cosh[x] + x*Sinh[x]

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Rubi [A]  time = 0.0177194, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ x \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]  Int[x*Cosh[x],x]

[Out]

-Cosh[x] + x*Sinh[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*cosh(x),x)

[Out]

x*sinh(x) - Integral(sinh(x), x)

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Mathematica [A]  time = 0.00497574, size = 9, normalized size = 1. \[ x \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Cosh[x],x]

[Out]

-Cosh[x] + x*Sinh[x]

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Maple [A]  time = 0.004, size = 10, normalized size = 1.1 \[ -\cosh \left ( x \right ) +x\sinh \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*cosh(x),x)

[Out]

-cosh(x)+x*sinh(x)

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Maxima [A]  time = 1.36487, size = 46, normalized size = 5.11 \[ \frac{1}{2} \, x^{2} \cosh \left (x\right ) - \frac{1}{4} \,{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac{1}{4} \,{\left (x^{2} - 2 \, x + 2\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cosh(x),x, algorithm="maxima")

[Out]

1/2*x^2*cosh(x) - 1/4*(x^2 + 2*x + 2)*e^(-x) - 1/4*(x^2 - 2*x + 2)*e^x

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Fricas [A]  time = 0.217322, size = 12, normalized size = 1.33 \[ x \sinh \left (x\right ) - \cosh \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cosh(x),x, algorithm="fricas")

[Out]

x*sinh(x) - cosh(x)

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Sympy [A]  time = 0.169484, size = 7, normalized size = 0.78 \[ x \sinh{\left (x \right )} - \cosh{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cosh(x),x)

[Out]

x*sinh(x) - cosh(x)

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GIAC/XCAS [A]  time = 0.228665, size = 23, normalized size = 2.56 \[ -\frac{1}{2} \,{\left (x + 1\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x - 1\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cosh(x),x, algorithm="giac")

[Out]

-1/2*(x + 1)*e^(-x) + 1/2*(x - 1)*e^x