3.3.27 \(\int \cosh (x) \cosh (4 x) \, dx\) [227]

Optimal. Leaf size=17 \[ \frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x) \]

[Out]

1/6*sinh(3*x)+1/10*sinh(5*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}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Cosh[4*x],x]

[Out]

Sinh[3*x]/6 + Sinh[5*x]/10

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 (4 x) \, dx &=\frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Cosh[4*x],x]

[Out]

Sinh[3*x]/6 + Sinh[5*x]/10

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Maple [A]
time = 0.78, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {\sinh \left (3 x \right )}{6}+\frac {\sinh \left (5 x \right )}{10}\)	\(14\)
risch	\(\frac {{\mathrm e}^{5 x}}{20}+\frac {{\mathrm e}^{3 x}}{12}-\frac {{\mathrm e}^{-3 x}}{12}-\frac {{\mathrm e}^{-5 x}}{20}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*cosh(4*x),x,method=_RETURNVERBOSE)

[Out]

1/6*sinh(3*x)+1/10*sinh(5*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 = 0.26, size = 27, normalized size = 1.59 \begin {gather*} \frac {1}{60} \, {\left (5 \, e^{\left (-2 \, x\right )} + 3\right )} e^{\left (5 \, x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} - \frac {1}{20} \, e^{\left (-5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x, algorithm="maxima")

[Out]

1/60*(5*e^(-2*x) + 3)*e^(5*x) - 1/12*e^(-3*x) - 1/20*e^(-5*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).
time = 0.37, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{10} \, \sinh \left (x\right )^{5} + \frac {1}{6} \, {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{3} + \frac {1}{2} \, {\left (\cosh \left (x\right )^{4} + \cosh \left (x\right )^{2}\right )} \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x, algorithm="fricas")

[Out]

1/10*sinh(x)^5 + 1/6*(6*cosh(x)^2 + 1)*sinh(x)^3 + 1/2*(cosh(x)^4 + cosh(x)^2)*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 (4 x \right )}}{15} + \frac {4 \sinh {\left (4 x \right )} \cosh {\left (x \right )}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x)

[Out]

-sinh(x)*cosh(4*x)/15 + 4*sinh(4*x)*cosh(x)/15

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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 = 0.38, size = 27, normalized size = 1.59 \begin {gather*} -\frac {1}{60} \, {\left (5 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{20} \, e^{\left (5 \, x\right )} + \frac {1}{12} \, e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x, algorithm="giac")

[Out]

-1/60*(5*e^(2*x) + 3)*e^(-5*x) + 1/20*e^(5*x) + 1/12*e^(3*x)

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Mupad [B]
time = 1.44, size = 15, normalized size = 0.88 \begin {gather*} \frac {8\,{\mathrm {sinh}\left (x\right )}^5}{5}+\frac {8\,{\mathrm {sinh}\left (x\right )}^3}{3}+\mathrm {sinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(4*x)*cosh(x),x)

[Out]

sinh(x) + (8*sinh(x)^3)/3 + (8*sinh(x)^5)/5

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