Integrand size = 13, antiderivative size = 110 \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {x}{2}+\frac {e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}} \]
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Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5485, 5483, 2266, 2235, 2236} \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {x}{2} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5483
Rule 5485
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2}+\frac {1}{2} \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{2} \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{4} \int e^{-2 a-2 b x-2 c x^2} \, dx+\frac {1}{4} \int e^{2 a+2 b x+2 c x^2} \, dx \\ & = \frac {x}{2}+\frac {1}{4} e^{2 a-\frac {b^2}{2 c}} \int e^{\frac {(2 b+4 c x)^2}{8 c}} \, dx+\frac {1}{4} e^{-2 a+\frac {b^2}{2 c}} \int e^{-\frac {(-2 b-4 c x)^2}{8 c}} \, dx \\ & = \frac {x}{2}+\frac {e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.27 \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {4 \sqrt {2} \sqrt {c} x+\sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+\sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )}{8 \sqrt {2} \sqrt {c}} \]
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Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {x}{2}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{8 \sqrt {-2 c}}\) | \(94\) |
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Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.16 \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } \sqrt {-c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {2} \sqrt {\pi } \sqrt {c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )}}{2 \, \sqrt {c}}\right ) - 8 \, c x}{16 \, c} \]
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\[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int \cosh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.87 \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x - \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{16 \, \sqrt {-c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x + \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{16 \, \sqrt {c}} + \frac {1}{2} \, x \]
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Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {c}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {-c}} + \frac {1}{2} \, x \]
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Timed out. \[ \int \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int {\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}^2 \,d x \]
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