3.300 \(\int x^5 \cosh (x) \, dx\)

Optimal. Leaf size=37 \[ x^5 \sinh (x)-5 x^4 \cosh (x)+20 x^3 \sinh (x)-60 x^2 \cosh (x)+120 x \sinh (x)-120 \cosh (x) \]

[Out]

-120*Cosh[x] - 60*x^2*Cosh[x] - 5*x^4*Cosh[x] + 120*x*Sinh[x] + 20*x^3*Sinh[x] +
x^5*Sinh[x]

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Rubi [A]  time = 0.116435, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x^5 \sinh (x)-5 x^4 \cosh (x)+20 x^3 \sinh (x)-60 x^2 \cosh (x)+120 x \sinh (x)-120 \cosh (x) \]

Antiderivative was successfully verified.

[In]  Int[x^5*Cosh[x],x]

[Out]

-120*Cosh[x] - 60*x^2*Cosh[x] - 5*x^4*Cosh[x] + 120*x*Sinh[x] + 20*x^3*Sinh[x] +
x^5*Sinh[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ x^{5} \sinh{\left (x \right )} - 5 x^{4} \cosh{\left (x \right )} + 20 x^{3} \sinh{\left (x \right )} - 60 x^{2} \cosh{\left (x \right )} + 120 x \sinh{\left (x \right )} - 120 \int \sinh{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*cosh(x),x)

[Out]

x**5*sinh(x) - 5*x**4*cosh(x) + 20*x**3*sinh(x) - 60*x**2*cosh(x) + 120*x*sinh(x
) - 120*Integral(sinh(x), x)

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Mathematica [A]  time = 0.0148117, size = 29, normalized size = 0.78 \[ x \left (x^4+20 x^2+120\right ) \sinh (x)-5 \left (x^4+12 x^2+24\right ) \cosh (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Cosh[x],x]

[Out]

-5*(24 + 12*x^2 + x^4)*Cosh[x] + x*(120 + 20*x^2 + x^4)*Sinh[x]

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Maple [A]  time = 0.01, size = 38, normalized size = 1. \[ -120\,\cosh \left ( x \right ) -60\,{x}^{2}\cosh \left ( x \right ) -5\,{x}^{4}\cosh \left ( x \right ) +120\,x\sinh \left ( x \right ) +20\,{x}^{3}\sinh \left ( x \right ) +{x}^{5}\sinh \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*cosh(x),x)

[Out]

-120*cosh(x)-60*x^2*cosh(x)-5*x^4*cosh(x)+120*x*sinh(x)+20*x^3*sinh(x)+x^5*sinh(
x)

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Maxima [A]  time = 1.33964, size = 100, normalized size = 2.7 \[ \frac{1}{6} \, x^{6} \cosh \left (x\right ) - \frac{1}{12} \,{\left (x^{6} + 6 \, x^{5} + 30 \, x^{4} + 120 \, x^{3} + 360 \, x^{2} + 720 \, x + 720\right )} e^{\left (-x\right )} - \frac{1}{12} \,{\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5*cosh(x),x, algorithm="maxima")

[Out]

1/6*x^6*cosh(x) - 1/12*(x^6 + 6*x^5 + 30*x^4 + 120*x^3 + 360*x^2 + 720*x + 720)*
e^(-x) - 1/12*(x^6 - 6*x^5 + 30*x^4 - 120*x^3 + 360*x^2 - 720*x + 720)*e^x

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Fricas [A]  time = 0.210484, size = 41, normalized size = 1.11 \[ -5 \,{\left (x^{4} + 12 \, x^{2} + 24\right )} \cosh \left (x\right ) +{\left (x^{5} + 20 \, x^{3} + 120 \, x\right )} \sinh \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5*cosh(x),x, algorithm="fricas")

[Out]

-5*(x^4 + 12*x^2 + 24)*cosh(x) + (x^5 + 20*x^3 + 120*x)*sinh(x)

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Sympy [A]  time = 3.09145, size = 42, normalized size = 1.14 \[ x^{5} \sinh{\left (x \right )} - 5 x^{4} \cosh{\left (x \right )} + 20 x^{3} \sinh{\left (x \right )} - 60 x^{2} \cosh{\left (x \right )} + 120 x \sinh{\left (x \right )} - 120 \cosh{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*cosh(x),x)

[Out]

x**5*sinh(x) - 5*x**4*cosh(x) + 20*x**3*sinh(x) - 60*x**2*cosh(x) + 120*x*sinh(x
) - 120*cosh(x)

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GIAC/XCAS [A]  time = 0.210449, size = 77, normalized size = 2.08 \[ -\frac{1}{2} \,{\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5*cosh(x),x, algorithm="giac")

[Out]

-1/2*(x^5 + 5*x^4 + 20*x^3 + 60*x^2 + 120*x + 120)*e^(-x) + 1/2*(x^5 - 5*x^4 + 2
0*x^3 - 60*x^2 + 120*x - 120)*e^x