3.590 \(\int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx\)

Optimal. Leaf size=30 \[ \frac{x}{4}+\frac{1}{8} \sinh (2 x)+\frac{1}{16} \sinh (4 x)+\frac{1}{24} \sinh (6 x) \]

[Out]

x/4 + Sinh[2*x]/8 + Sinh[4*x]/16 + Sinh[6*x]/24

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Rubi [A]  time = 0.0563583, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x}{4}+\frac{1}{8} \sinh (2 x)+\frac{1}{16} \sinh (4 x)+\frac{1}{24} \sinh (6 x) \]

Antiderivative was successfully verified.

[In]  Int[Cosh[x]*Cosh[2*x]*Cosh[3*x],x]

[Out]

x/4 + Sinh[2*x]/8 + Sinh[4*x]/16 + Sinh[6*x]/24

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\operatorname{atanh}{\left (\tanh{\left (3 x \right )} \right )}}{12} + \int \cosh{\left (2 x \right )}\, dx + \frac{\int \cosh{\left (4 x \right )}\, dx}{2} + \frac{\tanh{\left (3 x \right )}}{12 \left (- \tanh ^{2}{\left (3 x \right )} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cosh(x)*cosh(2*x)*cosh(3*x),x)

[Out]

atanh(tanh(3*x))/12 + Integral(cosh(2*x), x) + Integral(cosh(4*x), x)/2 + tanh(3
*x)/(12*(-tanh(3*x)**2 + 1))

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Mathematica [A]  time = 0.0153374, size = 30, normalized size = 1. \[ \frac{x}{4}+\frac{1}{8} \sinh (2 x)+\frac{1}{16} \sinh (4 x)+\frac{1}{24} \sinh (6 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cosh[x]*Cosh[2*x]*Cosh[3*x],x]

[Out]

x/4 + Sinh[2*x]/8 + Sinh[4*x]/16 + Sinh[6*x]/24

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Maple [A]  time = 0.058, size = 23, normalized size = 0.8 \[{\frac{x}{4}}+{\frac{\sinh \left ( 2\,x \right ) }{8}}+{\frac{\sinh \left ( 4\,x \right ) }{16}}+{\frac{\sinh \left ( 6\,x \right ) }{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cosh(x)*cosh(2*x)*cosh(3*x),x)

[Out]

1/4*x+1/8*sinh(2*x)+1/16*sinh(4*x)+1/24*sinh(6*x)

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Maxima [A]  time = 1.3573, size = 57, normalized size = 1.9 \[ \frac{1}{96} \,{\left (3 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 2\right )} e^{\left (6 \, x\right )} + \frac{1}{4} \, x - \frac{1}{16} \, e^{\left (-2 \, x\right )} - \frac{1}{32} \, e^{\left (-4 \, x\right )} - \frac{1}{48} \, e^{\left (-6 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(3*x)*cosh(2*x)*cosh(x),x, algorithm="maxima")

[Out]

1/96*(3*e^(-2*x) + 6*e^(-4*x) + 2)*e^(6*x) + 1/4*x - 1/16*e^(-2*x) - 1/32*e^(-4*
x) - 1/48*e^(-6*x)

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Fricas [A]  time = 0.209451, size = 59, normalized size = 1.97 \[ \frac{1}{4} \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \frac{1}{12} \,{\left (10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + \frac{1}{4} \,{\left (\cosh \left (x\right )^{5} + \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + \frac{1}{4} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(3*x)*cosh(2*x)*cosh(x),x, algorithm="fricas")

[Out]

1/4*cosh(x)*sinh(x)^5 + 1/12*(10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 1/4*(cosh(x)
^5 + cosh(x)^3 + cosh(x))*sinh(x) + 1/4*x

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Sympy [A]  time = 21.7791, size = 114, normalized size = 3.8 \[ \frac{x \sinh{\left (x \right )} \sinh{\left (2 x \right )} \cosh{\left (3 x \right )}}{4} - \frac{x \sinh{\left (x \right )} \sinh{\left (3 x \right )} \cosh{\left (2 x \right )}}{4} - \frac{x \sinh{\left (2 x \right )} \sinh{\left (3 x \right )} \cosh{\left (x \right )}}{4} + \frac{x \cosh{\left (x \right )} \cosh{\left (2 x \right )} \cosh{\left (3 x \right )}}{4} - \frac{\sinh{\left (x \right )} \cosh{\left (2 x \right )} \cosh{\left (3 x \right )}}{24} - \frac{\sinh{\left (2 x \right )} \cosh{\left (x \right )} \cosh{\left (3 x \right )}}{6} + \frac{3 \sinh{\left (3 x \right )} \cosh{\left (x \right )} \cosh{\left (2 x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)*cosh(2*x)*cosh(3*x),x)

[Out]

x*sinh(x)*sinh(2*x)*cosh(3*x)/4 - x*sinh(x)*sinh(3*x)*cosh(2*x)/4 - x*sinh(2*x)*
sinh(3*x)*cosh(x)/4 + x*cosh(x)*cosh(2*x)*cosh(3*x)/4 - sinh(x)*cosh(2*x)*cosh(3
*x)/24 - sinh(2*x)*cosh(x)*cosh(3*x)/6 + 3*sinh(3*x)*cosh(x)*cosh(2*x)/8

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GIAC/XCAS [A]  time = 0.199502, size = 65, normalized size = 2.17 \[ -\frac{1}{96} \,{\left (22 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-6 \, x\right )} + \frac{1}{4} \, x + \frac{1}{48} \, e^{\left (6 \, x\right )} + \frac{1}{32} \, e^{\left (4 \, x\right )} + \frac{1}{16} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(3*x)*cosh(2*x)*cosh(x),x, algorithm="giac")

[Out]

-1/96*(22*e^(6*x) + 6*e^(4*x) + 3*e^(2*x) + 2)*e^(-6*x) + 1/4*x + 1/48*e^(6*x) +
1/32*e^(4*x) + 1/16*e^(2*x)