Optimal. Leaf size=75 \[ -\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\frac{2 x+1}{\sqrt{2}}\right )+\frac{x^2}{4}+\frac{1}{8} \sinh \left (2 x^2+2 x+\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0541586, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {5395, 5383, 5375, 2234, 2204, 2205} \[ -\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\frac{2 x+1}{\sqrt{2}}\right )+\frac{x^2}{4}+\frac{1}{8} \sinh \left (2 x^2+2 x+\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 5395
Rule 5383
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \cosh ^2\left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \left (\frac{x}{2}+\frac{1}{2} x \cosh \left (\frac{1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac{x^2}{4}+\frac{1}{2} \int x \cosh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x^2}{4}+\frac{1}{8} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{4} \int \cosh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x^2}{4}+\frac{1}{8} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} \int e^{-\frac{1}{2}-2 x-2 x^2} \, dx-\frac{1}{8} \int e^{\frac{1}{2}+2 x+2 x^2} \, dx\\ &=\frac{x^2}{4}+\frac{1}{8} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} \int e^{-\frac{1}{8} (-2-4 x)^2} \, dx-\frac{1}{8} \int e^{\frac{1}{8} (2+4 x)^2} \, dx\\ &=\frac{x^2}{4}-\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{1+2 x}{\sqrt{2}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{1+2 x}{\sqrt{2}}\right )+\frac{1}{8} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.20472, size = 88, normalized size = 1.17 \[ \frac{-\sqrt{2 e \pi } \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )-\sqrt{2 e \pi } \text{Erfi}\left (\frac{2 x+1}{\sqrt{2}}\right )+8 \sqrt{e} x^2+2 (1+e) \sinh (2 x (x+1))+2 (e-1) \cosh (2 x (x+1))}{32 \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.046, size = 75, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{1}{16}{{\rm e}^{-{\frac{ \left ( 1+2\,x \right ) ^{2}}{2}}}}}-{\frac{\sqrt{\pi }\sqrt{2}}{32}{\it Erf} \left ( \sqrt{2}x+{\frac{\sqrt{2}}{2}} \right ) }+{\frac{1}{16}{{\rm e}^{{\frac{ \left ( 1+2\,x \right ) ^{2}}{2}}}}}+{\frac{i}{32}}\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 = 1.80211, size = 163, normalized size = 2.17 \begin{align*} \frac{1}{4} \, x^{2} - \frac{1}{32} \, \sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, x + 1\right )}{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt{-{\left (2 \, x + 1\right )}^{2}}} - \sqrt{2} e^{\left (\frac{1}{2} \,{\left (2 \, x + 1\right )}^{2}\right )}\right )} - \frac{1}{32} i \, \sqrt{2}{\left (-\frac{i \, \sqrt{\pi }{\left (2 \, x + 1\right )}{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt{{\left (2 \, x + 1\right )}^{2}}} - i \, \sqrt{2} e^{\left (-\frac{1}{2} \,{\left (2 \, x + 1\right )}^{2}\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27742, size = 263, normalized size = 3.51 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{2} e^{\left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )} - \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 )} e^{\left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )} + 2 \, e^{\left (4 \, x^{2} + 4 \, x + 1\right )} - 2\right )} e^{\left (-2 \, x^{2} - 2 \, x - \frac{1}{2}\right )} \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 x \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 = 1.23854, size = 95, normalized size = 1.27 \begin{align*} \frac{1}{4} \, x^{2} - \frac{1}{32} \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) - \frac{1}{32} i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} i \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + \frac{1}{16} \, e^{\left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )} - \frac{1}{16} \, e^{\left (-2 \, x^{2} - 2 \, x - \frac{1}{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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