Optimal. Leaf size=56 \[ \frac{1}{8} \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )+\frac{1}{8} \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\frac{2 x+1}{\sqrt{2}}\right )+\frac{x}{2} \]
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Rubi [A] time = 0.034158, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {5377, 5375, 2234, 2204, 2205} \[ \frac{1}{8} \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )+\frac{1}{8} \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\frac{2 x+1}{\sqrt{2}}\right )+\frac{x}{2} \]
Antiderivative was successfully verified.
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Rule 5377
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \cosh ^2\left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \left (\frac{1}{2}+\frac{1}{2} \cosh \left (\frac{1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac{x}{2}+\frac{1}{2} \int \cosh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x}{2}+\frac{1}{4} \int e^{-\frac{1}{2}-2 x-2 x^2} \, dx+\frac{1}{4} \int e^{\frac{1}{2}+2 x+2 x^2} \, dx\\ &=\frac{x}{2}+\frac{1}{4} \int e^{-\frac{1}{8} (-2-4 x)^2} \, dx+\frac{1}{4} \int e^{\frac{1}{8} (2+4 x)^2} \, dx\\ &=\frac{x}{2}+\frac{1}{8} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{1+2 x}{\sqrt{2}}\right )+\frac{1}{8} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{1+2 x}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0718669, size = 48, normalized size = 0.86 \[ \frac{1}{16} \left (\sqrt{2 \pi } \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )+\sqrt{2 \pi } \text{Erfi}\left (\frac{2 x+1}{\sqrt{2}}\right )+8 x\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.039, size = 49, normalized size = 0.9 \begin{align*}{\frac{x}{2}}+{\frac{\sqrt{\pi }\sqrt{2}}{16}{\it Erf} \left ( \sqrt{2}x+{\frac{\sqrt{2}}{2}} \right ) }-{\frac{i}{16}}\sqrt{\pi }\sqrt{2}{\it Erf} \left ( i\sqrt{2}x+{\frac{i}{2}}\sqrt{2} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.52938, size = 61, normalized size = 1.09 \begin{align*} \frac{1}{16} \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} x + \frac{1}{2} \, \sqrt{2}\right ) - \frac{1}{16} i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{2} x + \frac{1}{2} i \, \sqrt{2}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15691, size = 132, normalized size = 2.36 \begin{align*} \frac{1}{16} \, \sqrt{\pi }{\left (\sqrt{2} \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + \sqrt{2} \operatorname{erfi}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right )\right )} + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh ^{2}{\left (x^{2} + x + \frac{1}{4} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.24342, size = 57, normalized size = 1.02 \begin{align*} \frac{1}{16} \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + \frac{1}{16} i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} i \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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