3.276 \(\int x^2 \cosh (x) \, dx\)

Optimal. Leaf size=16 \[ x^2 \sinh (x)+2 \sinh (x)-2 x \cosh (x) \]

[Out]

-2*x*Cosh[x] + 2*Sinh[x] + x^2*Sinh[x]

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Rubi [A]  time = 0.0410497, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x^2 \sinh (x)+2 \sinh (x)-2 x \cosh (x) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Cosh[x],x]

[Out]

-2*x*Cosh[x] + 2*Sinh[x] + x^2*Sinh[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*cosh(x),x)

[Out]

x**2*sinh(x) - 2*x*cosh(x) + 2*Integral(cosh(x), x)

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Mathematica [A]  time = 0.00940846, size = 14, normalized size = 0.88 \[ \left (x^2+2\right ) \sinh (x)-2 x \cosh (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Cosh[x],x]

[Out]

-2*x*Cosh[x] + (2 + x^2)*Sinh[x]

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Maple [A]  time = 0.008, size = 17, normalized size = 1.1 \[ -2\,x\cosh \left ( x \right ) +2\,\sinh \left ( x \right ) +{x}^{2}\sinh \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*cosh(x),x)

[Out]

-2*x*cosh(x)+2*sinh(x)+x^2*sinh(x)

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Maxima [A]  time = 1.3776, size = 59, normalized size = 3.69 \[ \frac{1}{3} \, x^{3} \cosh \left (x\right ) - \frac{1}{6} \,{\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} - \frac{1}{6} \,{\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cosh(x),x, algorithm="maxima")

[Out]

1/3*x^3*cosh(x) - 1/6*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) - 1/6*(x^3 - 3*x^2 + 6*x -
6)*e^x

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Fricas [A]  time = 0.204867, size = 19, normalized size = 1.19 \[ -2 \, x \cosh \left (x\right ) +{\left (x^{2} + 2\right )} \sinh \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cosh(x),x, algorithm="fricas")

[Out]

-2*x*cosh(x) + (x^2 + 2)*sinh(x)

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Sympy [A]  time = 0.386933, size = 17, normalized size = 1.06 \[ x^{2} \sinh{\left (x \right )} - 2 x \cosh{\left (x \right )} + 2 \sinh{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*cosh(x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224227, size = 36, normalized size = 2.25 \[ -\frac{1}{2} \,{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x^{2} - 2 \, x + 2\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cosh(x),x, algorithm="giac")

[Out]

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