Integrand size = 14, antiderivative size = 100 \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=\frac {x}{2}+\frac {\sqrt {\pi } \cos \left (2 a+\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a+\frac {b^2}{2 c}\right )}{4 \sqrt {c}} \]
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Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3530, 3529, 3433, 3432} \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=\frac {\sqrt {\pi } \cos \left (2 a+\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } \sin \left (2 a+\frac {b^2}{2 c}\right ) \operatorname {FresnelS}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{4 \sqrt {c}}+\frac {x}{2} \]
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Rule 3432
Rule 3433
Rule 3529
Rule 3530
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (2 a+2 b x-2 c x^2\right )\right ) \, dx \\ & = \frac {x}{2}-\frac {1}{2} \int \cos \left (2 a+2 b x-2 c x^2\right ) \, dx \\ & = \frac {x}{2}-\frac {1}{2} \cos \left (2 a+\frac {b^2}{2 c}\right ) \int \cos \left (\frac {(2 b-4 c x)^2}{8 c}\right ) \, dx-\frac {1}{2} \sin \left (2 a+\frac {b^2}{2 c}\right ) \int \sin \left (\frac {(2 b-4 c x)^2}{8 c}\right ) \, dx \\ & = \frac {x}{2}+\frac {\sqrt {\pi } \cos \left (2 a+\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a+\frac {b^2}{2 c}\right )}{4 \sqrt {c}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=\frac {2 \sqrt {c} x-\sqrt {\pi } \cos \left (2 a+\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {-b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )-\sqrt {\pi } \operatorname {FresnelS}\left (\frac {-b+2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a+\frac {b^2}{2 c}\right )}{4 \sqrt {c}} \]
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Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {x}{2}-\frac {\sqrt {\pi }\, \left (\cos \left (\frac {4 a c +b^{2}}{2 c}\right ) \operatorname {C}\left (\frac {2 c x -b}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {4 a c +b^{2}}{2 c}\right ) \operatorname {S}\left (\frac {2 c x -b}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 \sqrt {c}}\) | \(76\) |
risch | \(\frac {x}{2}-\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c +b^{2}\right )}{2 c}} \operatorname {erf}\left (\sqrt {-2 i c}\, x +\frac {i b}{\sqrt {-2 i c}}\right )}{8 \sqrt {-2 i c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c +b^{2}\right )}{2 c}} \sqrt {2}\, \operatorname {erf}\left (-\sqrt {2}\, \sqrt {i c}\, x +\frac {i b \sqrt {2}}{2 \sqrt {i c}}\right )}{16 \sqrt {i c}}\) | \(107\) |
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.96 \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=-\frac {\pi \sqrt {\frac {c}{\pi }} \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) \operatorname {C}\left (\frac {{\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) + \pi \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {{\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - 2 \, c x}{4 \, c} \]
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Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=\frac {x}{2} - \frac {\sqrt {\pi } \sqrt {- \frac {1}{c}} \left (- \sin {\left (2 a + \frac {b^{2}}{2 c} \right )} S\left (\frac {2 b - 4 c x}{2 \sqrt {\pi } \sqrt {- c}}\right ) + \cos {\left (2 a + \frac {b^{2}}{2 c} \right )} C\left (\frac {2 b - 4 c x}{2 \sqrt {\pi } \sqrt {- c}}\right )\right )}{4} \]
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Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.24 \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=-\frac {4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + \left (i + 1\right ) \, \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x - i \, b}{\sqrt {2 i \, c}}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + \left (i - 1\right ) \, \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x - i \, b}{\sqrt {-2 i \, c}}\right )\right )} c^{\frac {3}{2}} - 16 \, c^{2} x}{32 \, c^{2}} \]
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Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.24 \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=\frac {1}{2} \, x - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{8 \, \sqrt {c} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {-i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{8 \, \sqrt {c} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}} \]
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Timed out. \[ \int \sin ^2\left (a+b x-c x^2\right ) \, dx=\int {\sin \left (-c\,x^2+b\,x+a\right )}^2 \,d x \]
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