Optimal. Leaf size=126 \[ \frac {\sqrt {\pi } b \cos \left (2 a+\frac {b^2}{2 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {\sqrt {\pi } b \sin \left (2 a+\frac {b^2}{2 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {\sin \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {x^2}{4} \]
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Rubi [A] time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3467, 3462, 3448, 3352, 3351} \[ \frac {\sqrt {\pi } b \cos \left (2 a+\frac {b^2}{2 c}\right ) \text {FresnelC}\left (\frac {b-2 c x}{\sqrt {\pi } \sqrt {c}}\right )}{8 c^{3/2}}+\frac {\sqrt {\pi } b \sin \left (2 a+\frac {b^2}{2 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {\sin \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {x^2}{4} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3448
Rule 3462
Rule 3467
Rubi steps
\begin {align*} \int x \sin ^2\left (a+b x-c x^2\right ) \, dx &=\int \left (\frac {x}{2}-\frac {1}{2} x \cos \left (2 a+2 b x-2 c x^2\right )\right ) \, dx\\ &=\frac {x^2}{4}-\frac {1}{2} \int x \cos \left (2 a+2 b x-2 c x^2\right ) \, dx\\ &=\frac {x^2}{4}+\frac {\sin \left (2 a+2 b x-2 c x^2\right )}{8 c}-\frac {b \int \cos \left (2 a+2 b x-2 c x^2\right ) \, dx}{4 c}\\ &=\frac {x^2}{4}+\frac {\sin \left (2 a+2 b x-2 c x^2\right )}{8 c}-\frac {\left (b \cos \left (2 a+\frac {b^2}{2 c}\right )\right ) \int \cos \left (\frac {(2 b-4 c x)^2}{8 c}\right ) \, dx}{4 c}-\frac {\left (b \sin \left (2 a+\frac {b^2}{2 c}\right )\right ) \int \sin \left (\frac {(2 b-4 c x)^2}{8 c}\right ) \, dx}{4 c}\\ &=\frac {x^2}{4}+\frac {b \sqrt {\pi } \cos \left (2 a+\frac {b^2}{2 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {b \sqrt {\pi } S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a+\frac {b^2}{2 c}\right )}{8 c^{3/2}}+\frac {\sin \left (2 a+2 b x-2 c x^2\right )}{8 c}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 122, normalized size = 0.97 \[ \frac {\sqrt {\pi } (-b) \cos \left (2 a+\frac {b^2}{2 c}\right ) C\left (\frac {2 c x-b}{\sqrt {c} \sqrt {\pi }}\right )-\sqrt {\pi } b \sin \left (2 a+\frac {b^2}{2 c}\right ) S\left (\frac {2 c x-b}{\sqrt {c} \sqrt {\pi }}\right )+\sqrt {c} \left (\sin (2 (a+x (b-c x)))+2 c x^2\right )}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 133, normalized size = 1.06 \[ -\frac {\pi b \sqrt {\frac {c}{\pi }} \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) \operatorname {C}\left (\frac {{\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) + \pi b \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {{\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - 2 \, c^{2} x^{2} + 2 \, c \cos \left (c x^{2} - b x - a\right ) \sin \left (c x^{2} - b x - a\right )}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.24, size = 172, normalized size = 1.37 \[ \frac {1}{4} \, x^{2} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{\sqrt {c} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}} + i \, e^{\left (2 i \, c x^{2} - 2 i \, b x - 2 i \, a\right )}}{16 \, c} + \frac {\frac {\sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {-i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{\sqrt {c} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}} - i \, e^{\left (-2 i \, c x^{2} + 2 i \, b x + 2 i \, a\right )}}{16 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 99, normalized size = 0.79 \[ \frac {x^{2}}{4}+\frac {\sin \left (-2 c \,x^{2}+2 b x +2 a \right )}{8 c}-\frac {b \sqrt {\pi }\, \left (\cos \left (\frac {4 c a +b^{2}}{2 c}\right ) \FresnelC \left (\frac {2 c x -b}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {4 c a +b^{2}}{2 c}\right ) \mathrm {S}\left (\frac {2 c x -b}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{8 c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.36, size = 608, normalized size = 4.83 \[ \frac {\sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left ({\left (\left (2 i - 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (-\left (2 i + 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} x + \sqrt {2} {\left (8 \, c^{2} x^{2} + c {\left (2 i \, e^{\left (\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} - 2 i \, e^{\left (-\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{2 \, c}\right )}\right )} \cos \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + 2 \, c {\left (e^{\left (\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} + e^{\left (-\frac {4 i \, c^{2} x^{2} - 4 i \, b c x + i \, b^{2}}{2 \, c}\right )}\right )} \sin \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sqrt {\frac {4 \, c^{2} x^{2} - 4 \, b c x + b^{2}}{c}}\right )}}{64 \, c^{2} \sqrt {\frac {4 \, c^{2} x^{2} - 4 \, b c x + b^{2}}{c}}} \]
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 \[ \int x\,{\sin \left (-c\,x^2+b\,x+a\right )}^2 \,d x \]
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 \[ \int x \sin ^{2}{\left (a + b x - c x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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