3.226 \(\int \cosh (x) \cosh (3 x) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{4} \sinh (2 x)+\frac {1}{8} \sinh (4 x) \]

[Out]

1/4*sinh(2*x)+1/8*sinh(4*x)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4283} \[ \frac {1}{4} \sinh (2 x)+\frac {1}{8} \sinh (4 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sinh[2*x]/4 + Sinh[4*x]/8

Rule 4283

Int[cos[(a_.) + (b_.)*(x_)]*cos[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
+ Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {1}{4} \sinh (2 x)+\frac {1}{8} \sinh (4 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sinh[2*x]/4 + Sinh[4*x]/8

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fricas [A]  time = 0.39, size = 20, normalized size = 1.18 \[ \frac {1}{2} \, \cosh \relax (x) \sinh \relax (x)^{3} + \frac {1}{2} \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(3*x),x, algorithm="fricas")

[Out]

1/2*cosh(x)*sinh(x)^3 + 1/2*(cosh(x)^3 + cosh(x))*sinh(x)

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giac [B]  time = 0.11, size = 27, normalized size = 1.59 \[ -\frac {1}{16} \, {\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} + \frac {1}{16} \, e^{\left (4 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(3*x),x, algorithm="giac")

[Out]

-1/16*(2*e^(2*x) + 1)*e^(-4*x) + 1/16*e^(4*x) + 1/8*e^(2*x)

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maple [A]  time = 0.21, size = 14, normalized size = 0.82 \[ \frac {\sinh \left (2 x \right )}{4}+\frac {\sinh \left (4 x \right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sinh(2*x)+1/8*sinh(4*x)

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maxima [B]  time = 0.33, size = 27, normalized size = 1.59 \[ \frac {1}{16} \, {\left (2 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{16} \, e^{\left (-4 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(3*x),x, algorithm="maxima")

[Out]

1/16*(2*e^(-2*x) + 1)*e^(4*x) - 1/8*e^(-2*x) - 1/16*e^(-4*x)

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mupad [B]  time = 1.45, size = 20, normalized size = 1.18 \[ \frac {{\mathrm {e}}^{-4\,x}\,\left ({\mathrm {e}}^{2\,x}-1\right )\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(-4*x)*(exp(2*x) - 1)*(exp(2*x) + 1)^3)/16

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sympy [A]  time = 0.41, size = 20, normalized size = 1.18 \[ - \frac {\sinh {\relax (x )} \cosh {\left (3 x \right )}}{8} + \frac {3 \sinh {\left (3 x \right )} \cosh {\relax (x )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(3*x),x)

[Out]

-sinh(x)*cosh(3*x)/8 + 3*sinh(3*x)*cosh(x)/8

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