Optimal. Leaf size=52 \[ \frac{1}{8} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )-\frac{1}{8} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right )+\frac{1}{2} \sinh \left (x^2+x+\frac{1}{4}\right ) \]
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Rubi [A] time = 0.025401, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {5383, 5375, 2234, 2204, 2205} \[ \frac{1}{8} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )-\frac{1}{8} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right )+\frac{1}{2} \sinh \left (x^2+x+\frac{1}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 5383
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int x \cosh \left (\frac{1}{4}+x+x^2\right ) \, dx &=\frac{1}{2} \sinh \left (\frac{1}{4}+x+x^2\right )-\frac{1}{2} \int \cosh \left (\frac{1}{4}+x+x^2\right ) \, dx\\ &=\frac{1}{2} \sinh \left (\frac{1}{4}+x+x^2\right )-\frac{1}{4} \int e^{-\frac{1}{4}-x-x^2} \, dx-\frac{1}{4} \int e^{\frac{1}{4}+x+x^2} \, dx\\ &=\frac{1}{2} \sinh \left (\frac{1}{4}+x+x^2\right )-\frac{1}{4} \int e^{-\frac{1}{4} (-1-2 x)^2} \, dx-\frac{1}{4} \int e^{\frac{1}{4} (1+2 x)^2} \, dx\\ &=\frac{1}{8} \sqrt{\pi } \text{erf}\left (\frac{1}{2} (-1-2 x)\right )-\frac{1}{8} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (1+2 x)\right )+\frac{1}{2} \sinh \left (\frac{1}{4}+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0737857, size = 76, normalized size = 1.46 \[ \frac{-\sqrt [4]{e} \sqrt{\pi } \text{Erf}\left (x+\frac{1}{2}\right )-\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (x+\frac{1}{2}\right )+2 \left (1+\sqrt{e}\right ) \sinh (x (x+1))+2 \left (\sqrt{e}-1\right ) \cosh (x (x+1))}{8 \sqrt [4]{e}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.028, size = 49, normalized size = 0.9 \begin{align*} -{\frac{1}{4}{{\rm e}^{-{\frac{ \left ( 1+2\,x \right ) ^{2}}{4}}}}}-{\frac{\sqrt{\pi }}{8}{\it Erf} \left ({\frac{1}{2}}+x \right ) }+{\frac{1}{4}{{\rm e}^{{\frac{ \left ( 1+2\,x \right ) ^{2}}{4}}}}}+{\frac{i}{8}}\sqrt{\pi }{\it Erf} \left ( ix+{\frac{i}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48487, size = 166, normalized size = 3.19 \begin{align*} \frac{1}{2} \, x^{2} \cosh \left (x^{2} + x + \frac{1}{4}\right ) + \frac{{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{4 \,{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac{3}{2}}} - \frac{{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{4 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac{3}{2}}} - \frac{1}{16} \, e^{\left (\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} - \frac{1}{16} \, e^{\left (-\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} - \frac{1}{4} \, \Gamma \left (2, \frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right ) + \frac{1}{4} \, \Gamma \left (2, -\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51533, size = 155, normalized size = 2.98 \begin{align*} -\frac{1}{8} \,{\left (\sqrt{\pi }{\left (\operatorname{erf}\left (x + \frac{1}{2}\right ) + \operatorname{erfi}\left (x + \frac{1}{2}\right )\right )} e^{\left (x^{2} + x + \frac{1}{4}\right )} - 2 \, e^{\left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )} + 2\right )} e^{\left (-x^{2} - x - \frac{1}{4}\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{\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 = 1.25733, size = 58, normalized size = 1.12 \begin{align*} -\frac{1}{8} \, \sqrt{\pi } \operatorname{erf}\left (x + \frac{1}{2}\right ) - \frac{1}{8} i \, \sqrt{\pi } \operatorname{erf}\left (-i \, x - \frac{1}{2} i\right ) + \frac{1}{4} \, e^{\left (x^{2} + x + \frac{1}{4}\right )} - \frac{1}{4} \, e^{\left (-x^{2} - x - \frac{1}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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