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

________________________________________________________________________________________

Rubi [A]  time = 0.0345891, 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, 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 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]

Rule 2637

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

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*}

Mathematica [A]  time = 0.0099438, 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

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 23, normalized size = 0.8 \begin{align*}{\frac{x}{4}}+{\frac{\sinh \left ( 2\,x \right ) }{8}}+{\frac{\sinh \left ( 4\,x \right ) }{16}}+{\frac{\sinh \left ( 6\,x \right ) }{24}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 0.951413, size = 57, normalized size = 1.9 \begin{align*} \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 )} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.08525, size = 166, normalized size = 5.53 \begin{align*} \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 \end{align*}

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

________________________________________________________________________________________

Sympy [B]  time = 13.0053, size = 116, normalized size = 3.87 \begin{align*} \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{3 \sinh{\left (x \right )} \sinh{\left (2 x \right )} \sinh{\left (3 x \right )}}{8} + \frac{\sinh{\left (x \right )} \cosh{\left (2 x \right )} \cosh{\left (3 x \right )}}{3} + \frac{5 \sinh{\left (2 x \right )} \cosh{\left (x \right )} \cosh{\left (3 x \right )}}{24} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.09158, size = 65, normalized size = 2.17 \begin{align*} -\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 )} \end{align*}

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)