Optimal. Leaf size=68 \[ \frac{1}{16} \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )-\frac{x^3}{6}+\frac{1}{8} x \sinh \left (2 x^2+2 x+\frac{1}{2}\right )-\frac{1}{16} \sinh \left (2 x^2+2 x+\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0959689, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {5394, 5387, 5374, 2234, 2204, 2205, 5383, 5375} \[ \frac{1}{16} \sqrt{\frac{\pi }{2}} \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )-\frac{x^3}{6}+\frac{1}{8} x \sinh \left (2 x^2+2 x+\frac{1}{2}\right )-\frac{1}{16} \sinh \left (2 x^2+2 x+\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 5394
Rule 5387
Rule 5374
Rule 2234
Rule 2204
Rule 2205
Rule 5383
Rule 5375
Rubi steps
\begin{align*} \int x^2 \sinh ^2\left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \left (-\frac{x^2}{2}+\frac{1}{2} x^2 \cosh \left (\frac{1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=-\frac{x^3}{6}+\frac{1}{2} \int x^2 \cosh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=-\frac{x^3}{6}+\frac{1}{8} x \sinh \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} \int \sinh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx-\frac{1}{4} \int x \cosh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=-\frac{x^3}{6}-\frac{1}{16} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{8} x \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{16} \int e^{-\frac{1}{2}-2 x-2 x^2} \, dx-\frac{1}{16} \int e^{\frac{1}{2}+2 x+2 x^2} \, dx+\frac{1}{8} \int \cosh \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=-\frac{x^3}{6}-\frac{1}{16} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{8} x \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{16} \int e^{-\frac{1}{8} (-2-4 x)^2} \, dx-\frac{1}{16} \int e^{\frac{1}{8} (2+4 x)^2} \, dx+\frac{1}{16} \int e^{-\frac{1}{2}-2 x-2 x^2} \, dx+\frac{1}{16} \int e^{\frac{1}{2}+2 x+2 x^2} \, dx\\ &=-\frac{x^3}{6}+\frac{1}{32} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{1+2 x}{\sqrt{2}}\right )-\frac{1}{32} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{1+2 x}{\sqrt{2}}\right )-\frac{1}{16} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{8} x \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{16} \int e^{-\frac{1}{8} (-2-4 x)^2} \, dx+\frac{1}{16} \int e^{\frac{1}{8} (2+4 x)^2} \, dx\\ &=-\frac{x^3}{6}+\frac{1}{16} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{1+2 x}{\sqrt{2}}\right )-\frac{1}{16} \sinh \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{8} x \sinh \left (\frac{1}{2}+2 x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.206504, size = 99, normalized size = 1.46 \[ \frac{3 \sqrt{2 e \pi } \text{Erf}\left (\frac{2 x+1}{\sqrt{2}}\right )-16 \sqrt{e} x^3+6 e x \sinh (2 x (x+1))+6 x \sinh (2 x (x+1))-3 e \sinh (2 x (x+1))-3 \sinh (2 x (x+1))+3 (e-1) (2 x-1) \cosh (2 x (x+1))}{96 \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 77, normalized size = 1.1 \begin{align*} -{\frac{{x}^{3}}{6}}-{\frac{x}{16}{{\rm e}^{-{\frac{ \left ( 1+2\,x \right ) ^{2}}{2}}}}}+{\frac{1}{32}{{\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{x}{16}{{\rm e}^{{\frac{ \left ( 1+2\,x \right ) ^{2}}{2}}}}}-{\frac{1}{32}{{\rm e}^{{\frac{ \left ( 1+2\,x \right ) ^{2}}{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.77605, size = 153, normalized size = 2.25 \begin{align*} -\frac{1}{6} \, x^{3} + \frac{1}{32} \,{\left (2 \, x e^{\frac{1}{2}} - e^{\frac{1}{2}}\right )} e^{\left (2 \, x^{2} + 2 \, x\right )} - \frac{1}{64} i \, \sqrt{2}{\left (-\frac{2 i \,{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{1}{2} \,{\left (2 \, x + 1\right )}^{2}\right )}{{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac{3}{2}}} + \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}}} + 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 [B] time = 2.0434, size = 860, normalized size = 12.65 \begin{align*} -\frac{16 \, x^{3} \cosh \left (x^{2} + x + \frac{1}{4}\right )^{2} - 3 \,{\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac{1}{4}\right )^{4} - 12 \,{\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac{1}{4}\right ) \sinh \left (x^{2} + x + \frac{1}{4}\right )^{3} - 3 \,{\left (2 \, x - 1\right )} \sinh \left (x^{2} + x + \frac{1}{4}\right )^{4} + 2 \,{\left (8 \, x^{3} - 9 \,{\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac{1}{4}\right )^{2}\right )} \sinh \left (x^{2} + x + \frac{1}{4}\right )^{2} + 4 \,{\left (8 \, x^{3} \cosh \left (x^{2} + x + \frac{1}{4}\right ) - 3 \,{\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac{1}{4}\right )^{3}\right )} \sinh \left (x^{2} + x + \frac{1}{4}\right ) - 3 \, \sqrt{\pi }{\left (\sqrt{2} \cosh \left (x^{2} + x + \frac{1}{4}\right )^{2} \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + 2 \, \sqrt{2} \cosh \left (x^{2} + x + \frac{1}{4}\right ) \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) \sinh \left (x^{2} + x + \frac{1}{4}\right ) + \sqrt{2} \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) \sinh \left (x^{2} + x + \frac{1}{4}\right )^{2}\right )} + 6 \, x - 3}{96 \,{\left (\cosh \left (x^{2} + x + \frac{1}{4}\right )^{2} + 2 \, \cosh \left (x^{2} + x + \frac{1}{4}\right ) \sinh \left (x^{2} + x + \frac{1}{4}\right ) + \sinh \left (x^{2} + x + \frac{1}{4}\right )^{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^{2} \sinh ^{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 [A] time = 1.32606, size = 82, normalized size = 1.21 \begin{align*} -\frac{1}{6} \, x^{3} + \frac{1}{32} \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + \frac{1}{32} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x^{2} + 2 \, x + \frac{1}{2}\right )} - \frac{1}{32} \,{\left (2 \, x - 1\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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