Optimal. Leaf size=91 \[ \frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]
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Rubi [A] time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5375, 2234, 2204, 2205} \[ \frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} \text {Erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 5375
Rubi steps
\begin {align*} \int \cosh \left (a+b x+c x^2\right ) \, dx &=\frac {1}{2} \int e^{-a-b x-c x^2} \, dx+\frac {1}{2} \int e^{a+b x+c x^2} \, dx\\ &=\frac {1}{2} e^{a-\frac {b^2}{4 c}} \int e^{\frac {(b+2 c x)^2}{4 c}} \, dx+\frac {1}{2} e^{-a+\frac {b^2}{4 c}} \int e^{-\frac {(-b-2 c x)^2}{4 c}} \, dx\\ &=\frac {e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 105, normalized size = 1.15 \[ \frac {\sqrt {\pi } \left (\text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )-\sinh \left (a-\frac {b^2}{4 c}\right )\right )+\text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\sinh \left (a-\frac {b^2}{4 c}\right )+\cosh \left (a-\frac {b^2}{4 c}\right )\right )\right )}{4 \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 112, normalized size = 1.23 \[ -\frac {\sqrt {\pi } \sqrt {-c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } \sqrt {c} {\left (\cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right )}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 79, normalized size = 0.87 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{4 \, \sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 83, normalized size = 0.91 \[ \frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}} \erf \left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}} \erf \left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{4 \sqrt {-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 464, normalized size = 5.10 \[ \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} b e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {c}} - \frac {1}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {5}{2}}} - \frac {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {2 \, c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} b e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {-c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {5}{2}}} + \frac {4 \, b c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} c e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {-c}} + x \cosh \left (c x^{2} + b x + a\right ) \]
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 \[ \int \mathrm {cosh}\left (c\,x^2+b\,x+a\right ) \,d x \]
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 \[ \int \cosh {\left (a + b x + c x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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