Optimal. Leaf size=183 \[ \frac{\sqrt{\pi } f^a e^{-\frac{e^2}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}+i d} \text{Erfi}\left (\frac{2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]
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Rubi [A] time = 0.304128, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4473, 2287, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } f^a e^{-\frac{e^2}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}+i d} \text{Erfi}\left (\frac{2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]
Antiderivative was successfully verified.
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Rule 4473
Rule 2287
Rule 2234
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int f^{a+c x^2} \cos \left (d+e x+f x^2\right ) \, dx &=\int \left (\frac{1}{2} e^{-i d-i e x-i f x^2} f^{a+c x^2}+\frac{1}{2} e^{i d+i e x+i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{2} \int e^{-i d-i e x-i f x^2} f^{a+c x^2} \, dx+\frac{1}{2} \int e^{i d+i e x+i f x^2} f^{a+c x^2} \, dx\\ &=\frac{1}{2} \int \exp \left (-i d-i e x+a \log (f)-x^2 (i f-c \log (f))\right ) \, dx+\frac{1}{2} \int \exp \left (i d+i e x+a \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{2} \left (e^{-i d-\frac{e^2}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-i e+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx+\frac{1}{2} \left (e^{i d+\frac{e^2}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(i e+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=\frac{e^{-i d-\frac{e^2}{4 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{i e+2 x (i f-c \log (f))}{2 \sqrt{i f-c \log (f)}}\right )}{4 \sqrt{i f-c \log (f)}}+\frac{e^{i d+\frac{e^2}{4 i f+4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+2 x (i f+c \log (f))}{2 \sqrt{i f+c \log (f)}}\right )}{4 \sqrt{i f+c \log (f)}}\\ \end{align*}
Mathematica [A] time = 0.941907, size = 217, normalized size = 1.19 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)+4 i f}} \left (\sqrt{f-i c \log (f)} (c \log (f)-i f) (\cos (d)+i \sin (d)) \text{Erfi}\left (\frac{\sqrt [4]{-1} (-2 i c x \log (f)+e+2 f x)}{2 \sqrt{f-i c \log (f)}}\right )-(f-i c \log (f)) \sqrt{f+i c \log (f)} (\cos (d)-i \sin (d)) e^{\frac{i e^2 f}{2 \left (c^2 \log ^2(f)+f^2\right )}} \text{Erfi}\left (\frac{(-1)^{3/4} (2 i c x \log (f)+e+2 f x)}{2 \sqrt{f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.09, size = 167, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,id\ln \left ( f \right ) c+4\,df-{e}^{2}}{4\,c\ln \left ( f \right ) -4\,if}}}}{\it Erf} \left ( x\sqrt{if-c\ln \left ( f \right ) }+{{\frac{i}{2}}e{\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}}-{\frac{{f}^{a}\sqrt{\pi }}{4}{{\rm e}^{{\frac{4\,id\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{{\frac{i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.548367, size = 765, normalized size = 4.18 \begin{align*} -\frac{\sqrt{\pi }{\left (c \log \left (f\right ) - i \, f\right )} \sqrt{-c \log \left (f\right ) - i \, f} \operatorname{erf}\left (\frac{{\left (2 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x + i \, c e \log \left (f\right ) + e f\right )} \sqrt{-c \log \left (f\right ) - i \, f}}{2 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (\frac{4 \, a c^{2} \log \left (f\right )^{3} + 4 i \, c^{2} d \log \left (f\right )^{2} - i \, e^{2} f + 4 i \, d f^{2} +{\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + \sqrt{\pi }{\left (c \log \left (f\right ) + i \, f\right )} \sqrt{-c \log \left (f\right ) + i \, f} \operatorname{erf}\left (\frac{{\left (2 \, c^{2} x \log \left (f\right )^{2} + 2 \, f^{2} x - i \, c e \log \left (f\right ) + e f\right )} \sqrt{-c \log \left (f\right ) + i \, f}}{2 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right ) e^{\left (\frac{4 \, a c^{2} \log \left (f\right )^{3} - 4 i \, c^{2} d \log \left (f\right )^{2} + i \, e^{2} f - 4 i \, d f^{2} +{\left (c e^{2} + 4 \, a f^{2}\right )} \log \left (f\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )}}{4 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + c x^{2}} \cos{\left (d + e x + f x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c x^{2} + a} \cos \left (f x^{2} + e x + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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