Optimal. Leaf size=110 \[ -\frac {\sqrt {\frac {\pi }{2}} e^{2 a+\frac {b^2}{2 c}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} e^{-2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {x}{2} \]
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Rubi [A] time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5377, 5375, 2234, 2205, 2204} \[ -\frac {\sqrt {\frac {\pi }{2}} e^{2 a+\frac {b^2}{2 c}} \text {Erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} e^{-2 a-\frac {b^2}{2 c}} \text {Erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{8 \sqrt {c}}+\frac {x}{2} \]
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 (a+b x-c x^2\right ) \, dx &=\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*}
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Mathematica [A] time = 0.15, size = 144, normalized size = 1.31 \[ \frac {\sqrt {\pi } \text {erf}\left (\frac {2 c x-b}{\sqrt {2} \sqrt {c}}\right ) \left (\sinh \left (2 a+\frac {b^2}{2 c}\right )+\cosh \left (2 a+\frac {b^2}{2 c}\right )\right )+\sqrt {\pi } \text {erfi}\left (\frac {2 c x-b}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a+\frac {b^2}{2 c}\right )-\sinh \left (2 a+\frac {b^2}{2 c}\right )\right )+4 \sqrt {2} \sqrt {c} x}{8 \sqrt {2} \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 132, normalized size = 1.20 \[ -\frac {\sqrt {2} \sqrt {\pi } \sqrt {-c} {\left (\cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {2} \sqrt {\pi } \sqrt {c} {\left (\cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )}}{2 \, \sqrt {c}}\right ) - 8 \, c x}{16 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 96, normalized size = 0.87 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {c}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} + 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {-c}} + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 90, normalized size = 0.82 \[ \frac {x}{2}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{2 c}} \erf \left (\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{8 \sqrt {-2 c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{2 c}} \sqrt {2}\, \erf \left (-\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 96, normalized size = 0.87 \[ \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x - \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (2 \, a + \frac {b^{2}}{2 \, c}\right )}}{16 \, \sqrt {c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x + \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (-2 \, a - \frac {b^{2}}{2 \, c}\right )}}{16 \, \sqrt {-c}} + \frac {1}{2} \, x \]
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 \[ \int {\mathrm {cosh}\left (-c\,x^2+b\,x+a\right )}^2 \,d x \]
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 \[ \int \cosh ^{2}{\left (a + b x - c x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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