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 = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {5394, 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 2204
Rule 2205
Rule 2234
Rule 5375
Rule 5383
Rule 5394
Rubi steps
\begin {align*} \int x \sinh ^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*}
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Mathematica [A] time = 0.21, 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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fricas [A] time = 0.65, size = 90, normalized size = 1.20 \[ -\frac {1}{32} \, {\left (8 \, x^{2} e^{\left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )} + \sqrt {\pi } {\left (\sqrt {2} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \sqrt {2} \operatorname {erfi}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right )\right )} e^{\left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )} - 2 \, e^{\left (4 \, x^{2} + 4 \, x + 1\right )} + 2\right )} e^{\left (-2 \, x^{2} - 2 \, x - \frac {1}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 70, normalized size = 0.93 \[ -\frac {1}{4} \, x^{2} - \frac {1}{32} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) - \frac {1}{32} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \frac {1}{16} \, e^{\left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )} - \frac {1}{16} \, e^{\left (-2 \, x^{2} - 2 \, x - \frac {1}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 75, normalized size = 1.00 \[ -\frac {x^{2}}{4}-\frac {{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{2}}}{16}-\frac {\sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, x +\frac {\sqrt {2}}{2}\right )}{32}+\frac {{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{2}}}{16}+\frac {i \sqrt {\pi }\, \sqrt {2}\, \erf \left (i \sqrt {2}\, x +\frac {i \sqrt {2}}{2}\right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.18, size = 121, normalized size = 1.61 \[ -\frac {1}{4} \, x^{2} - \frac {1}{32} \, \sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} - \sqrt {2} e^{\left (\frac {1}{2} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} - \frac {1}{32} i \, \sqrt {2} {\left (-\frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} - i \, \sqrt {2} e^{\left (-\frac {1}{2} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {sinh}\left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
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 \[ \int x \sinh ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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