Optimal. Leaf size=14 \[ \frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2635, 8} \[ \frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \cosh ^2(x) \, dx &=\frac {1}{2} \cosh (x) \sinh (x)+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}+\frac {1}{2} \cosh (x) \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {x}{2}+\frac {1}{4} \sinh (2 x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.12, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, \cosh \relax (x) \sinh \relax (x) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 24, normalized size = 1.71 \[ -\frac {1}{8} \, {\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} + \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {x}{2}+\frac {\cosh \relax (x ) \sinh \relax (x )}{2}\) | \(11\) |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{2 x}}{8}-\frac {{\mathrm e}^{-2 x}}{8}\) | \(17\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 16, normalized size = 1.14 \[ \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 10, normalized size = 0.71 \[ \frac {x}{2}+\frac {\mathrm {sinh}\left (2\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.20, size = 24, normalized size = 1.71 \[ - \frac {x \sinh ^{2}{\relax (x )}}{2} + \frac {x \cosh ^{2}{\relax (x )}}{2} + \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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