Optimal. Leaf size=56 \[ \frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {2 x+1}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {2 x+1}{\sqrt {2}}\right )+\frac {x}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5377, 5375, 2234, 2204, 2205} \[ \frac {1}{8} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {2 x+1}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {2 x+1}{\sqrt {2}}\right )+\frac {x}{2} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 5375
Rule 5377
Rubi steps
\begin {align*} \int \cosh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx &=\int \left (\frac {1}{2}+\frac {1}{2} \cosh \left (\frac {1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{2} \int \cosh \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{4} \int e^{-\frac {1}{2}-2 x-2 x^2} \, dx+\frac {1}{4} \int e^{\frac {1}{2}+2 x+2 x^2} \, dx\\ &=\frac {x}{2}+\frac {1}{4} \int e^{-\frac {1}{8} (-2-4 x)^2} \, dx+\frac {1}{4} \int e^{\frac {1}{8} (2+4 x)^2} \, dx\\ &=\frac {x}{2}+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {1+2 x}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {1+2 x}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 0.86 \[ \frac {1}{16} \left (\sqrt {2 \pi } \text {erf}\left (\frac {2 x+1}{\sqrt {2}}\right )+\sqrt {2 \pi } \text {erfi}\left (\frac {2 x+1}{\sqrt {2}}\right )+8 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 40, normalized size = 0.71 \[ \frac {1}{16} \, \sqrt {\pi } {\left (\sqrt {2} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \sqrt {2} \operatorname {erfi}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right )\right )} + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.13, size = 42, normalized size = 0.75 \[ \frac {1}{16} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \frac {1}{16} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 49, normalized size = 0.88 \[ \frac {x}{2}+\frac {\sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, x +\frac {\sqrt {2}}{2}\right )}{16}-\frac {i \sqrt {\pi }\, \sqrt {2}\, \erf \left (i \sqrt {2}\, x +\frac {i \sqrt {2}}{2}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 45, normalized size = 0.80 \[ \frac {1}{16} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x + \frac {1}{2} \, \sqrt {2}\right ) - \frac {1}{16} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {2} x + \frac {1}{2} i \, \sqrt {2}\right ) + \frac {1}{2} \, x \]
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 \[ \int {\mathrm {cosh}\left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
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 \[ \int \cosh ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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