Integrand size = 11, antiderivative size = 56 \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {x}{2}+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {1+2 x}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {1+2 x}{\sqrt {2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5484, 5483, 2266, 2235, 2236} \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {2 x+1}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {2 x+1}{\sqrt {2}}\right )-\frac {x}{2} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5483
Rule 5484
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{16} \left (-8 x+\sqrt {2 \pi } \text {erf}\left (\frac {1+2 x}{\sqrt {2}}\right )+\sqrt {2 \pi } \text {erfi}\left (\frac {1+2 x}{\sqrt {2}}\right )\right ) \]
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Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {x}{2}+\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x +\frac {\sqrt {2}}{2}\right )}{16}-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (i \sqrt {2}\, x +\frac {i \sqrt {2}}{2}\right )}{16}\) | \(49\) |
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none
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.73 \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{16} \, \sqrt {\pi } {\left (\sqrt {2} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) - \sqrt {-2} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {-2} {\left (2 \, x + 1\right )}\right )\right )} - \frac {1}{2} \, x \]
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\[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\int \sinh ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{16} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x + \frac {1}{2} \, \sqrt {2}\right ) - \frac {1}{16} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {2} x + \frac {1}{2} i \, \sqrt {2}\right ) - \frac {1}{2} \, x \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{16} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \frac {1}{16} i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) - \frac {1}{2} \, x \]
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Timed out. \[ \int \sinh ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\int {\mathrm {sinh}\left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
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