Optimal. Leaf size=39 \[ \frac{1}{4} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )+\frac{1}{4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right ) \]
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Rubi [A] time = 0.0146505, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {5374, 2234, 2204, 2205} \[ \frac{1}{4} \sqrt{\pi } \text{Erf}\left (\frac{1}{2} (-2 x-1)\right )+\frac{1}{4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} (2 x+1)\right ) \]
Antiderivative was successfully verified.
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Rule 5374
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sinh \left (\frac{1}{4}+x+x^2\right ) \, dx &=-\left (\frac{1}{2} \int e^{-\frac{1}{4}-x-x^2} \, dx\right )+\frac{1}{2} \int e^{\frac{1}{4}+x+x^2} \, dx\\ &=-\left (\frac{1}{2} \int e^{-\frac{1}{4} (-1-2 x)^2} \, dx\right )+\frac{1}{2} \int e^{\frac{1}{4} (1+2 x)^2} \, dx\\ &=\frac{1}{4} \sqrt{\pi } \text{erf}\left (\frac{1}{2} (-1-2 x)\right )+\frac{1}{4} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} (1+2 x)\right )\\ \end{align*}
Mathematica [A] time = 0.0245296, size = 24, normalized size = 0.62 \[ \frac{1}{4} \sqrt{\pi } \left (\text{Erfi}\left (x+\frac{1}{2}\right )-\text{Erf}\left (x+\frac{1}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 25, normalized size = 0.6 \begin{align*} -{\frac{\sqrt{\pi }}{4}{\it Erf} \left ({\frac{1}{2}}+x \right ) }-{\frac{i}{4}}\sqrt{\pi }{\it Erf} \left ( ix+{\frac{i}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.40334, size = 127, normalized size = 3.26 \begin{align*} \frac{{\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )}{2 \,{\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 )}{2 \, \left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac{3}{2}}} + x \sinh \left (x^{2} + x + \frac{1}{4}\right ) + \frac{1}{4} \, e^{\left (\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} - \frac{1}{4} \, e^{\left (-\frac{1}{4} \,{\left (2 \, x + 1\right )}^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07183, size = 62, normalized size = 1.59 \begin{align*} -\frac{1}{4} \, \sqrt{\pi }{\left (\operatorname{erf}\left (x + \frac{1}{2}\right ) - \operatorname{erfi}\left (x + \frac{1}{2}\right )\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 \sinh{\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 [C] time = 1.27667, size = 28, normalized size = 0.72 \begin{align*} -\frac{1}{4} \, \sqrt{\pi } \operatorname{erf}\left (x + \frac{1}{2}\right ) + \frac{1}{4} i \, \sqrt{\pi } \operatorname{erf}\left (-i \, x - \frac{1}{2} i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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