Integrand size = 18, antiderivative size = 103 \[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=\frac {e^{-i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}} \]
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Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4561, 2325, 2236, 2235} \[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } e^{-i d} f^a \text {erf}\left (x \sqrt {-c \log (f)+i f}\right )}{4 \sqrt {-c \log (f)+i f}}+\frac {\sqrt {\pi } e^{i d} f^a \text {erfi}\left (x \sqrt {c \log (f)+i f}\right )}{4 \sqrt {c \log (f)+i f}} \]
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Rule 2235
Rule 2236
Rule 2325
Rule 4561
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} e^{-i d-i f x^2} f^{a+c x^2}+\frac {1}{2} e^{i d+i f x^2} f^{a+c x^2}\right ) \, dx \\ & = \frac {1}{2} \int e^{-i d-i f x^2} f^{a+c x^2} \, dx+\frac {1}{2} \int e^{i d+i f x^2} f^{a+c x^2} \, dx \\ & = \frac {1}{2} \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx+\frac {1}{2} \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx \\ & = \frac {e^{-i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}+\frac {e^{i d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.65 \[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=-\frac {(-1)^{3/4} f^a \sqrt {\pi } \left (\text {erfi}\left (\sqrt [4]{-1} x \sqrt {f-i c \log (f)}\right ) \sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d))+\sqrt {f+i c \log (f)} \left (f \cos (d) \text {erf}\left (\frac {(1+i) x \sqrt {f+i c \log (f)}}{\sqrt {2}}\right )-\text {erfi}\left ((-1)^{3/4} x \sqrt {f+i c \log (f)}\right ) (c \cos (d) \log (f)+(f-i c \log (f)) \sin (d))\right )\right )}{4 \left (f^2+c^2 \log ^2(f)\right )} \]
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Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-i d} \operatorname {erf}\left (x \sqrt {i f -c \ln \left (f \right )}\right )}{4 \sqrt {i f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{i d} \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )-i f}\, x \right )}{4 \sqrt {-c \ln \left (f \right )-i f}}\) | \(82\) |
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none
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.06 \[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } {\left (c \log \left (f\right ) - i \, f\right )} \sqrt {-c \log \left (f\right ) - i \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right ) e^{\left (a \log \left (f\right ) + i \, d\right )} + \sqrt {\pi } {\left (c \log \left (f\right ) + i \, f\right )} \sqrt {-c \log \left (f\right ) + i \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) e^{\left (a \log \left (f\right ) - i \, d\right )}}{4 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]
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\[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=\int f^{a + c x^{2}} \cos {\left (d + f x^{2} \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (73) = 146\).
Time = 0.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.99 \[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left (f^{a} {\left (i \, \cos \left (d\right ) + \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) + f^{a} {\left (-i \, \cos \left (d\right ) + \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right )\right )} \sqrt {c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \left (f\right )^{2} + 2 \, f^{2}} {\left (f^{a} {\left (\cos \left (d\right ) - i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + i \, f} x\right ) + f^{a} {\left (\cos \left (d\right ) + i \, \sin \left (d\right )\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - i \, f} x\right )\right )} \sqrt {-c \log \left (f\right ) + \sqrt {c^{2} \log \left (f\right )^{2} + f^{2}}}}{8 \, {\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \]
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\[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=\int { f^{c x^{2} + a} \cos \left (f x^{2} + d\right ) \,d x } \]
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Timed out. \[ \int f^{a+c x^2} \cos \left (d+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,\cos \left (f\,x^2+d\right ) \,d x \]
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