Optimal. Leaf size=14 \[ \frac{x}{2}+\frac{1}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.0071077, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{x}{2}+\frac{1}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0018328, size = 14, normalized size = 1. \[ \frac{x}{2}+\frac{1}{4} \sinh (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}}+{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926173, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{2} \, x + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03191, size = 39, normalized size = 2.79 \begin{align*} \frac{1}{2} \, \cosh \left (x\right ) \sinh \left (x\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.184481, size = 24, normalized size = 1.71 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09885, size = 32, normalized size = 2.29 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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