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 = 0.10, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 2204
Rule 2205
Rule 2234
Rule 5374
Rule 5375
Rule 5383
Rule 5387
Rule 5394
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*}
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Mathematica [A] time = 0.21, 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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fricas [B] time = 0.93, size = 268, normalized size = 3.94 \[ -\frac {16 \, x^{3} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{2} - 3 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{4} - 12 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right ) \sinh \left (x^{2} + x + \frac {1}{4}\right )^{3} - 3 \, {\left (2 \, x - 1\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right )^{4} + 2 \, {\left (8 \, x^{3} - 9 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{2}\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right )^{2} + 4 \, {\left (8 \, x^{3} \cosh \left (x^{2} + x + \frac {1}{4}\right ) - 3 \, {\left (2 \, x - 1\right )} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{3}\right )} \sinh \left (x^{2} + x + \frac {1}{4}\right ) - 3 \, \sqrt {\pi } {\left (\sqrt {2} \cosh \left (x^{2} + x + \frac {1}{4}\right )^{2} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + 2 \, \sqrt {2} \cosh \left (x^{2} + x + \frac {1}{4}\right ) \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) \sinh \left (x^{2} + x + \frac {1}{4}\right ) + \sqrt {2} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) \sinh \left (x^{2} + x + \frac {1}{4}\right )^{2}\right )} + 6 \, x - 3}{96 \, {\left (\cosh \left (x^{2} + x + \frac {1}{4}\right )^{2} + 2 \, \cosh \left (x^{2} + x + \frac {1}{4}\right ) \sinh \left (x^{2} + x + \frac {1}{4}\right ) + \sinh \left (x^{2} + x + \frac {1}{4}\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 61, normalized size = 0.90 \[ -\frac {1}{6} \, x^{3} + \frac {1}{32} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \frac {1}{32} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )} - \frac {1}{32} \, {\left (2 \, x - 1\right )} e^{\left (-2 \, x^{2} - 2 \, x - \frac {1}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 77, normalized size = 1.13 \[ -\frac {x^{3}}{6}-\frac {x \,{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{2}}}{16}+\frac {{\mathrm e}^{-\frac {\left (1+2 x \right )^{2}}{2}}}{32}+\frac {\sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, x +\frac {\sqrt {2}}{2}\right )}{32}+\frac {x \,{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{2}}}{16}-\frac {{\mathrm e}^{\frac {\left (1+2 x \right )^{2}}{2}}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.13, size = 113, normalized size = 1.66 \[ -\frac {1}{6} \, x^{3} + \frac {1}{32} \, {\left (2 \, x e^{\frac {1}{2}} - e^{\frac {1}{2}}\right )} e^{\left (2 \, x^{2} + 2 \, x\right )} - \frac {1}{64} i \, \sqrt {2} {\left (-\frac {2 i \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{2} \, {\left (2 \, x + 1\right )}^{2}\right )}{{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} + 2 i \, \sqrt {2} e^{\left (-\frac {1}{2} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} \]
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 \[ \int x^2\,{\mathrm {sinh}\left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
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 \[ \int x^{2} \sinh ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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