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

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

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4355, 2637} \[ \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \]

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4355

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.)*(H_)[(e_.) + (f_.)*(x_)]^(r_.), x_Symbol] :>
Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p
, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin {align*} \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx &=\int \left (\frac {1}{4}+\frac {1}{4} \cosh (2 x)+\frac {1}{4} \cosh (4 x)+\frac {1}{4} \cosh (6 x)\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{4} \int \cosh (2 x) \, dx+\frac {1}{4} \int \cosh (4 x) \, dx+\frac {1}{4} \int \cosh (6 x) \, dx\\ &=\frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \[ \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cosh[x]*Cosh[2*x]*Cosh[3*x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.01, size = 44, normalized size = 1.47 \[ \frac {1}{4} \, \cosh \relax (x) \sinh \relax (x)^{5} + \frac {1}{12} \, {\left (10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + \frac {1}{4} \, {\left (\cosh \relax (x)^{5} + \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + \frac {1}{4} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(2*x)*cosh(3*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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giac [B]  time = 0.63, size = 48, normalized size = 1.60 \[ -\frac {1}{96} \, {\left (22 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{4} \, x + \frac {1}{48} \, e^{\left (6 \, x\right )} + \frac {1}{32} \, e^{\left (4 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(2*x)*cosh(3*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)

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maple [A]  time = 0.13, size = 23, normalized size = 0.77

method	result	size
default	\(\frac {x}{4}+\frac {\sinh \left (2 x \right )}{8}+\frac {\sinh \left (4 x \right )}{16}+\frac {\sinh \left (6 x \right )}{24}\)	\(23\)
risch	\(\frac {x}{4}+\frac {{\mathrm e}^{6 x}}{48}+\frac {{\mathrm e}^{4 x}}{32}+\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{-2 x}}{16}-\frac {{\mathrm e}^{-4 x}}{32}-\frac {{\mathrm e}^{-6 x}}{48}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*cosh(2*x)*cosh(3*x),x,method=_RETURNVERBOSE)

[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 = 0.44, size = 42, normalized size = 1.40 \[ \frac {1}{96} \, {\left (3 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 2\right )} e^{\left (6 \, x\right )} + \frac {1}{4} \, x - \frac {1}{16} \, e^{\left (-2 \, x\right )} - \frac {1}{32} \, e^{\left (-4 \, x\right )} - \frac {1}{48} \, e^{\left (-6 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(2*x)*cosh(3*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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mupad [B]  time = 0.40, size = 40, normalized size = 1.33 \[ \frac {x}{4}-\frac {{\mathrm {e}}^{-2\,x}}{16}+\frac {{\mathrm {e}}^{2\,x}}{16}-\frac {{\mathrm {e}}^{-4\,x}}{32}+\frac {{\mathrm {e}}^{4\,x}}{32}-\frac {{\mathrm {e}}^{-6\,x}}{48}+\frac {{\mathrm {e}}^{6\,x}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(2*x)*cosh(3*x)*cosh(x),x)

[Out]

x/4 - exp(-2*x)/16 + exp(2*x)/16 - exp(-4*x)/32 + exp(4*x)/32 - exp(-6*x)/48 + exp(6*x)/48

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sympy [B]  time = 13.12, size = 116, normalized size = 3.87 \[ \frac {x \sinh {\relax (x )} \sinh {\left (2 x \right )} \cosh {\left (3 x \right )}}{4} - \frac {x \sinh {\relax (x )} \sinh {\left (3 x \right )} \cosh {\left (2 x \right )}}{4} - \frac {x \sinh {\left (2 x \right )} \sinh {\left (3 x \right )} \cosh {\relax (x )}}{4} + \frac {x \cosh {\relax (x )} \cosh {\left (2 x \right )} \cosh {\left (3 x \right )}}{4} - \frac {3 \sinh {\relax (x )} \sinh {\left (2 x \right )} \sinh {\left (3 x \right )}}{8} + \frac {\sinh {\relax (x )} \cosh {\left (2 x \right )} \cosh {\left (3 x \right )}}{3} + \frac {5 \sinh {\left (2 x \right )} \cosh {\relax (x )} \cosh {\left (3 x \right )}}{24} \]

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 - 3*sinh(x)*sinh(2*x)*sinh(3*x)/8 + sinh(x)*cosh(2*x)*cosh(3*x)/3 + 5*sinh(2*x)*cosh(x
)*cosh(3*x)/24

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