Optimal. Leaf size=97 \[ \frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}} \]
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Rubi [A] time = 0.02, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3447, 3351, 3352} \[ \frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \text {FresnelC}\left (\frac {b+2 c x}{\sqrt {2 \pi } \sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3447
Rubi steps
\begin {align*} \int \sin \left (a+b x+c x^2\right ) \, dx &=\cos \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right ) \, dx+\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right ) \, dx\\ &=\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 84, normalized size = 0.87 \[ \frac {\sqrt {\frac {\pi }{2}} \left (\sin \left (a-\frac {b^2}{4 c}\right ) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+\cos \left (a-\frac {b^2}{4 c}\right ) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 102, normalized size = 1.05 \[ \frac {\sqrt {2} \pi \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} \pi \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.19, size = 135, normalized size = 1.39 \[ \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{4 \, {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{4 \, {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 82, normalized size = 0.85 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-c a}{c}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-c a}{c}\right ) \FresnelC \left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.33, size = 112, normalized size = 1.15 \[ -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \left (i - 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{2 \, \sqrt {i \, c}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \left (i + 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{2 \, \sqrt {-i \, c}}\right )\right )}}{8 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 99, normalized size = 1.02 \[ \frac {\sqrt {2}\,\sqrt {\pi }\,\cos \left (\frac {4\,a\,c-b^2}{4\,c}\right )\,\mathrm {S}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}+c\,x\right )\,\sqrt {\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{c}}}{2}+\frac {\sqrt {2}\,\sqrt {\pi }\,\sin \left (\frac {4\,a\,c-b^2}{4\,c}\right )\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}+c\,x\right )\,\sqrt {\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{c}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 88, normalized size = 0.91 \[ \frac {\sqrt {2} \sqrt {\pi } \left (\sin {\left (a - \frac {b^{2}}{4 c} \right )} C\left (\frac {\sqrt {2} \left (b + 2 c x\right )}{2 \sqrt {\pi } \sqrt {c}}\right ) + \cos {\left (a - \frac {b^{2}}{4 c} \right )} S\left (\frac {\sqrt {2} \left (b + 2 c x\right )}{2 \sqrt {\pi } \sqrt {c}}\right )\right ) \sqrt {\frac {1}{c}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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