Integrand size = 21, antiderivative size = 179 \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\left (\frac {1}{16}+\frac {i}{16}\right ) e^{2 i d+\frac {i (2 i e+b \log (f))^2}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (2 i e+4 i f x+b \log (f))}{\sqrt {f}}\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) e^{-2 i d+\frac {i (2 e+i b \log (f))^2}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (2 i e+4 i f x-b \log (f))}{\sqrt {f}}\right )+\frac {f^{a+b x}}{2 b \log (f)} \]
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Time = 0.39 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4560, 2225, 2325, 2266, 2235, 2236} \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {i (b \log (f)+2 i e)^2}{8 f}+2 i d} \text {erf}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (b \log (f)+2 i e+4 i f x)}{\sqrt {f}}\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {i (2 e+i b \log (f))^2}{8 f}-2 i d} \text {erfi}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (-b \log (f)+2 i e+4 i f x)}{\sqrt {f}}\right )+\frac {f^{a+b x}}{2 b \log (f)} \]
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Rule 2225
Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 4560
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} f^{a+b x}-\frac {1}{4} e^{-2 i d-2 i e x-2 i f x^2} f^{a+b x}-\frac {1}{4} e^{2 i d+2 i e x+2 i f x^2} f^{a+b x}\right ) \, dx \\ & = -\left (\frac {1}{4} \int e^{-2 i d-2 i e x-2 i f x^2} f^{a+b x} \, dx\right )-\frac {1}{4} \int e^{2 i d+2 i e x+2 i f x^2} f^{a+b x} \, dx+\frac {1}{2} \int f^{a+b x} \, dx \\ & = \frac {f^{a+b x}}{2 b \log (f)}-\frac {1}{4} \int \exp \left (-2 i d-2 i f x^2+a \log (f)-x (2 i e-b \log (f))\right ) \, dx-\frac {1}{4} \int \exp \left (2 i d+2 i f x^2+a \log (f)+x (2 i e+b \log (f))\right ) \, dx \\ & = \frac {f^{a+b x}}{2 b \log (f)}-\frac {1}{4} \exp \left (-2 i d+a \log (f)-\frac {i (-2 i e+b \log (f))^2}{8 f}\right ) \int e^{\frac {i (-2 i e-4 i f x+b \log (f))^2}{8 f}} \, dx-\frac {1}{4} \left (e^{2 i d+\frac {i (2 i e+b \log (f))^2}{8 f}} f^a\right ) \int e^{-\frac {i (2 i e+4 i f x+b \log (f))^2}{8 f}} \, dx \\ & = \left (\frac {1}{16}+\frac {i}{16}\right ) e^{2 i d+\frac {i (2 i e+b \log (f))^2}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (2 i e+4 i f x+b \log (f))}{\sqrt {f}}\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) \exp \left (-\frac {1}{8} i \left (16 d+\frac {(2 i e-b \log (f))^2}{f}\right )\right ) f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (2 i e+4 i f x-b \log (f))}{\sqrt {f}}\right )+\frac {f^{a+b x}}{2 b \log (f)} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.36 \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {e^{-\frac {i \left (4 e^2+b^2 \log ^2(f)\right )}{8 f}} f^{a-\frac {b e+f}{2 f}} \left (8 e^{\frac {i \left (4 e^2+b^2 \log ^2(f)\right )}{8 f}} f^{\frac {1}{2}+b \left (\frac {e}{2 f}+x\right )}+\sqrt [4]{-1} b e^{\frac {i b^2 \log ^2(f)}{4 f}} \sqrt {2 \pi } \text {erf}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (2 i (e+2 f x)+b \log (f))}{\sqrt {f}}\right ) \log (f) (\cos (2 d)+i \sin (2 d))+\sqrt [4]{-1} b e^{\frac {i e^2}{f}} \sqrt {2 \pi } \text {erf}\left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (2 e+4 f x+i b \log (f))}{\sqrt {f}}\right ) \log (f) (i \cos (2 d)+\sin (2 d))\right )}{16 b \log (f)} \]
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Time = 0.90 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b e}{2 f}} {\mathrm e}^{-\frac {i \left (\ln \left (f \right )^{2} b^{2}+16 d f -4 e^{2}\right )}{8 f}} \sqrt {2}\, \operatorname {erf}\left (-\sqrt {2}\, \sqrt {i f}\, x +\frac {\left (b \ln \left (f \right )-2 i e \right ) \sqrt {2}}{4 \sqrt {i f}}\right )}{16 \sqrt {i f}}+\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b e}{2 f}} {\mathrm e}^{\frac {i \left (\ln \left (f \right )^{2} b^{2}+16 d f -4 e^{2}\right )}{8 f}} \operatorname {erf}\left (-\sqrt {-2 i f}\, x +\frac {2 i e +b \ln \left (f \right )}{2 \sqrt {-2 i f}}\right )}{8 \sqrt {-2 i f}}+\frac {f^{x b +a}}{2 b \ln \left (f \right )}\) | \(179\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (116) = 232\).
Time = 0.26 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.82 \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=-\frac {\pi b \sqrt {\frac {f}{\pi }} e^{\left (\frac {-i \, b^{2} \log \left (f\right )^{2} + 4 i \, e^{2} - 16 i \, d f - 4 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{8 \, f}\right )} \operatorname {C}\left (\frac {{\left (4 \, f x + i \, b \log \left (f\right ) + 2 \, e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) \log \left (f\right ) - \pi b \sqrt {\frac {f}{\pi }} e^{\left (\frac {i \, b^{2} \log \left (f\right )^{2} - 4 i \, e^{2} + 16 i \, d f - 4 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{8 \, f}\right )} \operatorname {C}\left (-\frac {{\left (4 \, f x - i \, b \log \left (f\right ) + 2 \, e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) \log \left (f\right ) - i \, \pi b \sqrt {\frac {f}{\pi }} e^{\left (\frac {-i \, b^{2} \log \left (f\right )^{2} + 4 i \, e^{2} - 16 i \, d f - 4 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{8 \, f}\right )} \operatorname {S}\left (\frac {{\left (4 \, f x + i \, b \log \left (f\right ) + 2 \, e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) \log \left (f\right ) - i \, \pi b \sqrt {\frac {f}{\pi }} e^{\left (\frac {i \, b^{2} \log \left (f\right )^{2} - 4 i \, e^{2} + 16 i \, d f - 4 \, {\left (b e - 2 \, a f\right )} \log \left (f\right )}{8 \, f}\right )} \operatorname {S}\left (-\frac {{\left (4 \, f x - i \, b \log \left (f\right ) + 2 \, e\right )} \sqrt {\frac {f}{\pi }}}{2 \, f}\right ) \log \left (f\right ) - 4 \, f f^{b x + a}}{8 \, b f \log \left (f\right )} \]
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\[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int f^{a + b x} \sin ^{2}{\left (d + e x + f x^{2} \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (116) = 232\).
Time = 0.33 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.34 \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\frac {4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, b f^{a} \cos \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, e^{2} + 16 \, d f}{8 \, f}\right ) \log \left (f\right ) - \left (i + 1\right ) \, b f^{a} \log \left (f\right ) \sin \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, e^{2} + 16 \, d f}{8 \, f}\right )\right )} \operatorname {erf}\left (\frac {i \, {\left (4 i \, f x - b \log \left (f\right ) + 2 i \, e\right )} \sqrt {2 i \, f}}{4 \, f}\right ) + {\left (\left (i + 1\right ) \, b f^{a} \cos \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, e^{2} + 16 \, d f}{8 \, f}\right ) \log \left (f\right ) + \left (i - 1\right ) \, b f^{a} \log \left (f\right ) \sin \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, e^{2} + 16 \, d f}{8 \, f}\right )\right )} \operatorname {erf}\left (\frac {i \, {\left (4 i \, f x + b \log \left (f\right ) + 2 i \, e\right )} \sqrt {-2 i \, f}}{4 \, f}\right )\right )} f^{\frac {3}{2}} + 16 \, f^{a + 2} e^{\left (b x \log \left (f\right ) + \frac {b e \log \left (f\right )}{2 \, f}\right )}}{32 \, b f^{2} f^{\frac {b e}{2 \, f}} \log \left (f\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (116) = 232\).
Time = 0.37 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.35 \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int f^{a+b x} \sin ^2\left (d+e x+f x^2\right ) \, dx=\int f^{a+b\,x}\,{\sin \left (f\,x^2+e\,x+d\right )}^2 \,d x \]
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