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 = 0.02, 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 = {4440, 2717}
\begin {gather*} \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 4440
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 = 0.02, size = 30, normalized size = 1.00 \begin {gather*} \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, 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.
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Maxima [A]
time = 2.63, size = 42, normalized size = 1.40 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 44, normalized size = 1.47 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (22) = 44\).
time = 1.20, size = 114, normalized size = 3.80 \begin {gather*} \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 {\sinh {\left (x \right )} \cosh {\left (2 x \right )} \cosh {\left (3 x \right )}}{24} - \frac {\sinh {\left (2 x \right )} \cosh {\left (x \right )} \cosh {\left (3 x \right )}}{6} + \frac {3 \sinh {\left (3 x \right )} \cosh {\left (x \right )} \cosh {\left (2 x \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (22) = 44\).
time = 1.51, size = 48, normalized size = 1.60 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 40, normalized size = 1.33 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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