Optimal. Leaf size=66 \[ \frac{3}{16} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )-\frac{1}{16} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right )+\frac{1}{2} x \cosh \left (x^2+x+\frac{1}{4}\right )-\frac{1}{4} \cosh \left (x^2+x+\frac{1}{4}\right ) \]
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Rubi [A] time = 0.056489, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {5386, 5375, 2234, 2204, 2205, 5382, 5374} \[ \frac{3}{16} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )-\frac{1}{16} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right )+\frac{1}{2} x \cosh \left (x^2+x+\frac{1}{4}\right )-\frac{1}{4} \cosh \left (x^2+x+\frac{1}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 5386
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rule 5382
Rule 5374
Rubi steps
\begin{align*} \int x^2 \sinh \left (\frac{1}{4}+x+x^2\right ) \, dx &=\frac{1}{2} x \cosh \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} \int x \sinh \left (\frac{1}{4}+x+x^2\right ) \, dx\\ &=-\frac{1}{4} \cosh \left (\frac{1}{4}+x+x^2\right )+\frac{1}{2} x \cosh \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}{4} \int \sinh \left (\frac{1}{4}+x+x^2\right ) \, dx\\ &=-\frac{1}{4} \cosh \left (\frac{1}{4}+x+x^2\right )+\frac{1}{2} x \cosh \left (\frac{1}{4}+x+x^2\right )-\frac{1}{8} \int e^{-\frac{1}{4}-x-x^2} \, dx+\frac{1}{8} \int e^{\frac{1}{4}+x+x^2} \, dx-\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}{4} \cosh \left (\frac{1}{4}+x+x^2\right )+\frac{1}{2} x \cosh \left (\frac{1}{4}+x+x^2\right )+\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}{8} \int e^{-\frac{1}{4} (-1-2 x)^2} \, dx+\frac{1}{8} \int e^{\frac{1}{4} (1+2 x)^2} \, dx\\ &=-\frac{1}{4} \cosh \left (\frac{1}{4}+x+x^2\right )+\frac{1}{2} x \cosh \left (\frac{1}{4}+x+x^2\right )+\frac{3}{16} \sqrt{\pi } \text{erf}\left (\frac{1}{2} (-1-2 x)\right )-\frac{1}{16} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (1+2 x)\right )\\ \end{align*}
Mathematica [A] time = 0.150503, size = 72, normalized size = 1.09 \[ \frac{1}{16} \left (-3 \sqrt{\pi } \text{Erf}\left (x+\frac{1}{2}\right )-\sqrt{\pi } \text{Erfi}\left (x+\frac{1}{2}\right )+\frac{2 (2 x-1) \left (\left (\sqrt{e}-1\right ) \sinh (x (x+1))+\left (1+\sqrt{e}\right ) \cosh (x (x+1))\right )}{\sqrt [4]{e}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.04, size = 75, normalized size = 1.1 \begin{align*}{\frac{x}{4}{{\rm e}^{-{\frac{ \left ( 1+2\,x \right ) ^{2}}{4}}}}}-{\frac{1}{8}{{\rm e}^{-{\frac{ \left ( 1+2\,x \right ) ^{2}}{4}}}}}-{\frac{3\,\sqrt{\pi }}{16}{\it Erf} \left ({\frac{1}{2}}+x \right ) }+{\frac{x}{4}{{\rm e}^{{\frac{ \left ( 1+2\,x \right ) ^{2}}{4}}}}}-{\frac{1}{8}{{\rm e}^{{\frac{ \left ( 1+2\,x \right ) ^{2}}{4}}}}}+{\frac{i}{16}}\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 = 1.43015, size = 247, normalized size = 3.74 \begin{align*} \frac{1}{3} \, x^{3} \sinh \left (x^{2} + x + \frac{1}{4}\right ) + \frac{{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac{5}{2}, \frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{6 \,{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac{5}{2}}} + \frac{{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac{5}{2}, -\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{6 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac{5}{2}}} + \frac{{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{8 \,{\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 )}{8 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac{3}{2}}} + \frac{1}{48} \, e^{\left (\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} - \frac{1}{48} \, 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 = 2.17879, size = 181, normalized size = 2.74 \begin{align*} -\frac{1}{16} \,{\left (\sqrt{\pi }{\left (3 \, \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 \,{\left (2 \, x - 1\right )} e^{\left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )} - 4 \, x + 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^{2} \sinh{\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 = 1.39717, size = 72, normalized size = 1.09 \begin{align*} \frac{1}{8} \,{\left (2 \, x - 1\right )} e^{\left (x^{2} + x + \frac{1}{4}\right )} + \frac{1}{8} \,{\left (2 \, x - 1\right )} e^{\left (-x^{2} - x - \frac{1}{4}\right )} - \frac{3}{16} \, \sqrt{\pi } \operatorname{erf}\left (x + \frac{1}{2}\right ) - \frac{1}{16} i \, \sqrt{\pi } \operatorname{erf}\left (-i \, x - \frac{1}{2} i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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