Optimal. Leaf size=14 \[ \frac {x}{2}+\frac {1}{2} \cosh (x) \sinh (x) \]
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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 = {2715, 8}
\begin {gather*} \frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
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 \begin {gather*} \frac {x}{2}+\frac {1}{4} \sinh (2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {x}{2}+\frac {\cosh \left (x \right ) \sinh \left (x \right )}{2}\) | \(11\) |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{2 x}}{8}-\frac {{\mathrm e}^{-2 x}}{8}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.49, size = 16, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \cosh \left (x\right ) \sinh \left (x\right ) + \frac {1}{2} \, 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. 24 vs.
\(2 (10) = 20\).
time = 0.05, size = 24, normalized size = 1.71 \begin {gather*} - \frac {x \sinh ^{2}{\left (x \right )}}{2} + \frac {x \cosh ^{2}{\left (x \right )}}{2} + \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{2} \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. 24 vs.
\(2 (10) = 20\).
time = 0.95, size = 24, normalized size = 1.71 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \frac {x}{2}+\frac {\mathrm {sinh}\left (2\,x\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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