Integrand size = 24, antiderivative size = 209 \[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=\frac {e^{-\left ((i-\log (f)) \left (a-\frac {b^2 (i-\log (f))}{4 i e-4 c \log (f)}\right )\right )} \sqrt {\pi } \text {erf}\left (\frac {b (i-\log (f))+2 x (i e-c \log (f))}{2 \sqrt {i e-c \log (f)}}\right )}{4 \sqrt {i e-c \log (f)}}+\frac {e^{(i+\log (f)) \left (a-\frac {b^2 (i+\log (f))}{4 i e+4 c \log (f)}\right )} \sqrt {\pi } \text {erfi}\left (\frac {b (i+\log (f))+2 x (i e+c \log (f))}{2 \sqrt {i e+c \log (f)}}\right )}{4 \sqrt {i e+c \log (f)}} \]
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Time = 0.83 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4561, 2325, 2266, 2236, 2235} \[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=\frac {\sqrt {\pi } \exp \left (-\left ((-\log (f)+i) \left (a-\frac {b^2 (-\log (f)+i)}{-4 c \log (f)+4 i e}\right )\right )\right ) \text {erf}\left (\frac {b (-\log (f)+i)+2 x (-c \log (f)+i e)}{2 \sqrt {-c \log (f)+i e}}\right )}{4 \sqrt {-c \log (f)+i e}}+\frac {\sqrt {\pi } \exp \left ((\log (f)+i) \left (a-\frac {b^2 (\log (f)+i)}{4 c \log (f)+4 i e}\right )\right ) \text {erfi}\left (\frac {b (\log (f)+i)+2 x (c \log (f)+i e)}{2 \sqrt {c \log (f)+i e}}\right )}{4 \sqrt {c \log (f)+i e}} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 4561
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} e^{-i a-i b x-i e x^2} f^{a+b x+c x^2}+\frac {1}{2} e^{i a+i b x+i e x^2} f^{a+b x+c x^2}\right ) \, dx \\ & = \frac {1}{2} \int e^{-i a-i b x-i e x^2} f^{a+b x+c x^2} \, dx+\frac {1}{2} \int e^{i a+i b x+i e x^2} f^{a+b x+c x^2} \, dx \\ & = \frac {1}{2} \int \exp \left (-a (i-\log (f))-b x (i-\log (f))-x^2 (i e-c \log (f))\right ) \, dx+\frac {1}{2} \int \exp \left (a (i+\log (f))+b x (i+\log (f))+x^2 (i e+c \log (f))\right ) \, dx \\ & = \frac {1}{2} \exp \left (-\left ((i-\log (f)) \left (a-\frac {b^2 (i-\log (f))}{4 i e-4 c \log (f)}\right )\right )\right ) \int \exp \left (\frac {(-b (i-\log (f))+2 x (-i e+c \log (f)))^2}{4 (-i e+c \log (f))}\right ) \, dx+\frac {1}{2} \exp \left ((i+\log (f)) \left (a-\frac {b^2 (i+\log (f))}{4 i e+4 c \log (f)}\right )\right ) \int \exp \left (\frac {(b (i+\log (f))+2 x (i e+c \log (f)))^2}{4 (i e+c \log (f))}\right ) \, dx \\ & = \frac {\exp \left (-\left ((i-\log (f)) \left (a-\frac {b^2 (i-\log (f))}{4 i e-4 c \log (f)}\right )\right )\right ) \sqrt {\pi } \text {erf}\left (\frac {b (i-\log (f))+2 x (i e-c \log (f))}{2 \sqrt {i e-c \log (f)}}\right )}{4 \sqrt {i e-c \log (f)}}+\frac {\exp \left ((i+\log (f)) \left (a-\frac {b^2 (i+\log (f))}{4 i e+4 c \log (f)}\right )\right ) \sqrt {\pi } \text {erfi}\left (\frac {b (i+\log (f))+2 x (i e+c \log (f))}{2 \sqrt {i e+c \log (f)}}\right )}{4 \sqrt {i e+c \log (f)}} \\ \end{align*}
Time = 1.64 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.56 \[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=-\frac {i e^{-\frac {b^2 c \log ^3(f)}{2 \left (e^2+c^2 \log ^2(f)\right )}} f^{a-\frac {b^2}{2 (e-i c \log (f))}} \sqrt {\pi } \left (-e^{\frac {1}{4} b^2 \left (\frac {1}{-i e+c \log (f)}+\frac {\log ^2(f)}{i e+c \log (f)}\right )} f^{\frac {i b^2 c \log (f)}{e^2+c^2 \log ^2(f)}} \text {erfi}\left (\frac {-i (b+2 e x)+(b+2 c x) \log (f)}{2 \sqrt {-i e+c \log (f)}}\right ) (e-i c \log (f)) \sqrt {-i e+c \log (f)} (\cos (a)-i \sin (a))+e^{\frac {1}{4} b^2 \left (\frac {\log ^2(f)}{-i e+c \log (f)}+\frac {1}{i e+c \log (f)}\right )} \text {erfi}\left (\frac {i (b+2 e x)+(b+2 c x) \log (f)}{2 \sqrt {i e+c \log (f)}}\right ) (e+i c \log (f)) \sqrt {i e+c \log (f)} (\cos (a)+i \sin (a))\right )}{4 \left (e^2+c^2 \log ^2(f)\right )} \]
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Time = 0.56 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \left (f \right )^{2} b^{2}+4 i \ln \left (f \right ) a c -2 i \ln \left (f \right ) b^{2}+4 a e -b^{2}}{4 \left (c \ln \left (f \right )-i e \right )}} \operatorname {erf}\left (-\sqrt {i e -c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-i b}{2 \sqrt {i e -c \ln \left (f \right )}}\right )}{4 \sqrt {i e -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {-\ln \left (f \right )^{2} b^{2}+4 i \ln \left (f \right ) a c -2 i \ln \left (f \right ) b^{2}-4 a e +b^{2}}{4 i e +4 c \ln \left (f \right )}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-i e}\, x +\frac {b \ln \left (f \right )+i b}{2 \sqrt {-c \ln \left (f \right )-i e}}\right )}{4 \sqrt {-c \ln \left (f \right )-i e}}\) | \(215\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (153) = 306\).
Time = 0.27 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.82 \[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=-\frac {\sqrt {\pi } {\left (c \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right ) - i \, e} \operatorname {erf}\left (\frac {{\left (2 \, e^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + b e + {\left (i \, b c - i \, b e\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - i \, e}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + i \, b^{2} e - 4 i \, a e^{2} - {\left (-2 i \, b^{2} c + 4 i \, a c^{2} + i \, b^{2} e\right )} \log \left (f\right )^{2} - {\left (b^{2} c - 2 \, b^{2} e + 4 \, a e^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )} + \sqrt {\pi } {\left (c \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right ) + i \, e} \operatorname {erf}\left (\frac {{\left (2 \, e^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + b e + {\left (-i \, b c + i \, b e\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + i \, e}}{2 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - i \, b^{2} e + 4 i \, a e^{2} - {\left (2 i \, b^{2} c - 4 i \, a c^{2} - i \, b^{2} e\right )} \log \left (f\right )^{2} - {\left (b^{2} c - 2 \, b^{2} e + 4 \, a e^{2}\right )} \log \left (f\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + e^{2}\right )}} \]
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\[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=\int f^{a + b x + c x^{2}} \cos {\left (a + b x + e x^{2} \right )}\, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.27 (sec) , antiderivative size = 1058, normalized size of antiderivative = 5.06 \[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=\text {Too large to display} \]
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\[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=\int { f^{c x^{2} + b x + a} \cos \left (e x^{2} + b x + a\right ) \,d x } \]
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Timed out. \[ \int f^{a+b x+c x^2} \cos \left (a+b x+e x^2\right ) \, dx=\int f^{c\,x^2+b\,x+a}\,\cos \left (e\,x^2+b\,x+a\right ) \,d x \]
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