Integrand size = 16, antiderivative size = 147 \[ \int f^{a+c x^2} \cos (d+e x) \, dx=-\frac {e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4561, 2325, 2266, 2235} \[ \int f^{a+c x^2} \cos (d+e x) \, dx=\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}+i d} \text {erfi}\left (\frac {2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 c \log (f)}-i d} \text {erfi}\left (\frac {-2 c x \log (f)+i e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2266
Rule 2325
Rule 4561
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} e^{-i d-i e x} f^{a+c x^2}+\frac {1}{2} e^{i d+i e x} f^{a+c x^2}\right ) \, dx \\ & = \frac {1}{2} \int e^{-i d-i e x} f^{a+c x^2} \, dx+\frac {1}{2} \int e^{i d+i e x} f^{a+c x^2} \, dx \\ & = \frac {1}{2} \int e^{-i d-i e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{2} \int e^{i d+i e x+a \log (f)+c x^2 \log (f)} \, dx \\ & = \frac {1}{2} \left (e^{-i d+\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{2} \left (e^{i d+\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx \\ & = -\frac {e^{-i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{i d+\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79 \[ \int f^{a+c x^2} \cos (d+e x) \, dx=\frac {e^{\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left (\text {erfi}\left (\frac {-i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)-i \sin (d))+\text {erfi}\left (\frac {i e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cos (d)+i \sin (d))\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.41 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {i e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(121\) |
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none
Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int f^{a+c x^2} \cos (d+e x) \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )} + \sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - i \, e\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 i \, c d \log \left (f\right ) + e^{2}}{4 \, c \log \left (f\right )}\right )}}{4 \, c \log \left (f\right )} \]
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\[ \int f^{a+c x^2} \cos (d+e x) \, dx=\int f^{a + c x^{2}} \cos {\left (d + e x \right )}\, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.39 \[ \int f^{a+c x^2} \cos (d+e x) \, dx=-\frac {\sqrt {\pi } {\left (f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} + \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} + f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \left (f\right )}} - \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \left (f\right )}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) + i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )} - f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\frac {2 \, c x \log \left (f\right ) - i \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \left (f\right )}\right )}\right )} \sqrt {-c \log \left (f\right )}}{8 \, c \log \left (f\right )} \]
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\[ \int f^{a+c x^2} \cos (d+e x) \, dx=\int { f^{c x^{2} + a} \cos \left (e x + d\right ) \,d x } \]
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Timed out. \[ \int f^{a+c x^2} \cos (d+e x) \, dx=\int f^{c\,x^2+a}\,\cos \left (d+e\,x\right ) \,d x \]
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