3.3.26 \(\int \cosh (x) \cosh (3 x) \, dx\) [226]

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 = {4368} \begin {gather*} \frac {1}{4} \sinh (2 x)+\frac {1}{8} \sinh (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sinh[2*x]/4 + Sinh[4*x]/8

Rule 4368

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 \begin {gather*} \frac {1}{4} \sinh (2 x)+\frac {1}{8} \sinh (4 x) \end {gather*}

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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Maple [A]
time = 0.97, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {\sinh \left (2 x \right )}{4}+\frac {\sinh \left (4 x \right )}{8}\)	\(14\)
risch	\(\frac {{\mathrm e}^{4 x}}{16}+\frac {{\mathrm e}^{2 x}}{8}-\frac {{\mathrm e}^{-2 x}}{8}-\frac {{\mathrm e}^{-4 x}}{16}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
time = 0.27, size = 27, normalized size = 1.59 \begin {gather*} \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 )} \end {gather*}

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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Fricas [A]
time = 0.35, size = 20, normalized size = 1.18 \begin {gather*} \frac {1}{2} \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \frac {1}{2} \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) \end {gather*}

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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Sympy [A]
time = 0.12, size = 20, normalized size = 1.18 \begin {gather*} - \frac {\sinh {\left (x \right )} \cosh {\left (3 x \right )}}{8} + \frac {3 \sinh {\left (3 x \right )} \cosh {\left (x \right )}}{8} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
time = 0.39, size = 27, normalized size = 1.59 \begin {gather*} -\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 )} \end {gather*}

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

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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