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 = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 2204
Rule 2205
Rule 2234
Rule 5374
Rule 5375
Rule 5382
Rule 5386
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*}
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Mathematica [A] time = 0.15, 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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fricas [A] time = 0.56, size = 58, normalized size = 0.88 \[ -\frac {1}{16} \, {\left (\sqrt {\pi } {\left (3 \, \operatorname {erf}\left (x + \frac {1}{2}\right ) + \operatorname {erfi}\left (x + \frac {1}{2}\right )\right )} e^{\left (x^{2} + x + \frac {1}{4}\right )} - 2 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )} - 4 \, x + 2\right )} e^{\left (-x^{2} - x - \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.15, size = 53, normalized size = 0.80 \[ \frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (x^{2} + x + \frac {1}{4}\right )} + \frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (-x^{2} - x - \frac {1}{4}\right )} - \frac {3}{16} \, \sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) - \frac {1}{16} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, x - \frac {1}{2} i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 75, normalized size = 1.14 \[ \frac {x \,{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{4}}}{4}-\frac {{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{4}}}{8}-\frac {3 \erf \left (\frac {1}{2}+x \right ) \sqrt {\pi }}{16}+\frac {x \,{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{4}}}{4}-\frac {{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{4}}}{8}+\frac {i \sqrt {\pi }\, \erf \left (i x +\frac {1}{2} i\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 183, normalized size = 2.77 \[ \frac {1}{3} \, x^{3} \sinh \left (x^{2} + x + \frac {1}{4}\right ) + \frac {{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac {5}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{6 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (2 \, x + 1\right )}^{5} \Gamma \left (\frac {5}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{6 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {5}{2}}} + \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{8 \, {\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{8 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} + \frac {1}{48} \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} - \frac {1}{48} \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} - \frac {1}{4} \, \Gamma \left (2, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right ) - \frac {1}{4} \, \Gamma \left (2, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\mathrm {sinh}\left (x^2+x+\frac {1}{4}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sinh {\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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