Integrand size = 19, antiderivative size = 150 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^2}{4 e}-\frac {(2 c d-b e) \sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {(2 c d-b e) \sqrt {\pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {b^2}{2 c}\right )}{8 c^{3/2}}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c} \]
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Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3548, 3543, 3529, 3433, 3432} \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}+\frac {\sqrt {\pi } \sin \left (2 a-\frac {b^2}{2 c}\right ) (2 c d-b e) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )}{8 c^{3/2}}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(d+e x)^2}{4 e} \]
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Rule 3432
Rule 3433
Rule 3529
Rule 3543
Rule 3548
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} (d+e x)-\frac {1}{2} (d+e x) \cos \left (2 a+2 b x+2 c x^2\right )\right ) \, dx \\ & = \frac {(d+e x)^2}{4 e}-\frac {1}{2} \int (d+e x) \cos \left (2 a+2 b x+2 c x^2\right ) \, dx \\ & = \frac {(d+e x)^2}{4 e}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {(4 c d-2 b e) \int \cos \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c} \\ & = \frac {(d+e x)^2}{4 e}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {\left ((2 c d-b e) \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 ((2 c d-b e) \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 {(d+e x)^2}{4 e}-\frac {(2 c d-b e) \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 )}{8 c^{3/2}}+\frac {(2 c d-b e) \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 )}{8 c^{3/2}}-\frac {e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {-\left ((2 c d-b e) \sqrt {\pi } \cos \left (2 a-\frac {b^2}{2 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right )\right )+(2 c d-b e) \sqrt {\pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {b^2}{2 c}\right )+\sqrt {c} (2 c x (2 d+e x)-e \sin (2 (a+x (b+c x))))}{8 c^{3/2}} \]
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Time = 0.44 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.13
method | result | size |
default | \(-\frac {e \sin \left (2 c \,x^{2}+2 b x +2 a \right )}{8 c}+\frac {e b \sqrt {\pi }\, \left (\cos \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {C}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )+\sin \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {S}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )\right )}{8 c^{\frac {3}{2}}}-\frac {\sqrt {\pi }\, d \left (\cos \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {C}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )+\sin \left (\frac {-4 a c +b^{2}}{2 c}\right ) \operatorname {S}\left (\frac {2 c x +b}{\sqrt {c}\, \sqrt {\pi }}\right )\right )}{4 \sqrt {c}}+\frac {d x}{2}+\frac {e \,x^{2}}{4}\) | \(170\) |
risch | \(-\frac {\operatorname {erf}\left (\sqrt {2}\, \sqrt {i c}\, x +\frac {i b \sqrt {2}}{2 \sqrt {i c}}\right ) \sqrt {2}\, \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{16 \sqrt {i c}}+\frac {e \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {i c}\, x +\frac {i b \sqrt {2}}{2 \sqrt {i c}}\right ) \sqrt {2}\, \sqrt {\pi }\, b \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{32 \sqrt {i c}\, c}+\frac {\operatorname {erf}\left (-\sqrt {-2 i c}\, x +\frac {i b}{\sqrt {-2 i c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{8 \sqrt {-2 i c}}-\frac {e \,\operatorname {erf}\left (-\sqrt {-2 i c}\, x +\frac {i b}{\sqrt {-2 i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{2 c}}}{16 \sqrt {-2 i c}\, c}+\frac {d x}{2}+\frac {e \,x^{2}}{4}-\frac {e \sin \left (2 c \,x^{2}+2 b x +2 a \right )}{8 c}\) | \(257\) |
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Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {2 \, c^{2} e x^{2} - \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) \operatorname {C}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) + \pi {\left (2 \, c d - b e\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + 4 \, c^{2} d x - 2 \, c e \cos \left (c x^{2} + b x + a\right ) \sin \left (c x^{2} + b x + a\right )}{8 \, c^{2}} \]
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\[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \sin ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 738, normalized size of antiderivative = 4.92 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {{\left (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\right )} d}{32 \, c^{2}} + \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 ) - 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 c \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 c \sin \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} x + 2 \, \sqrt {2} {\left (4 \, c^{2} x^{2} - c {\left (-i \, e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{2 \, c}\right )} + 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 ) - 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 )} e}{64 \, c^{2} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.30 \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, e x^{2} + \frac {1}{2} \, d x - \frac {-i \, e e^{\left (2 i \, c x^{2} + 2 i \, b x + 2 i \, a\right )} - \frac {\sqrt {\pi } {\left (-2 i \, c d + i \, b e\right )} \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{\sqrt {c} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )}}}{16 \, c} - \frac {i \, e e^{\left (-2 i \, c x^{2} - 2 i \, b x - 2 i \, a\right )} - \frac {\sqrt {\pi } {\left (2 i \, c d - i \, b e\right )} \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{\sqrt {c} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )}}}{16 \, c} \]
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Timed out. \[ \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx=\int {\sin \left (c\,x^2+b\,x+a\right )}^2\,\left (d+e\,x\right ) \,d x \]
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