Optimal. Leaf size=208 \[ \frac{\sqrt{\pi } f^a \exp \left (-\frac{(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } f^a \exp \left (\frac{(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]
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Rubi [A] time = 0.394717, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4473, 2287, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } f^a \exp \left (-\frac{(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}+\frac{\sqrt{\pi } f^a \exp \left (\frac{(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]
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+b x+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+b x+c x^2}+\frac{1}{2} e^{i d+i e x+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac{1}{2} \int e^{-i d-i e x-i f x^2} f^{a+b x+c x^2} \, dx+\frac{1}{2} \int e^{i d+i e x+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac{1}{2} \int \exp \left (-i d+a \log (f)-x (i e-b \log (f))-x^2 (i f-c \log (f))\right ) \, dx+\frac{1}{2} \int \exp \left (i d+a \log (f)+x (i e+b \log (f))+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{2} \left (\exp \left (-i d-\frac{(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac{(-i e+b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx+\frac{1}{2} \left (\exp \left (i d+\frac{(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac{(i e+b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=\frac{\exp \left (-i d-\frac{(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a \sqrt{\pi } \text{erf}\left (\frac{i e-b \log (f)+2 x (i f-c \log (f))}{2 \sqrt{i f-c \log (f)}}\right )}{4 \sqrt{i f-c \log (f)}}+\frac{\exp \left (i d+\frac{(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+b \log (f)+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 = 2.12557, size = 348, normalized size = 1.67 \[ \frac{\sqrt [4]{-1} \sqrt{\pi } f^{\frac{f (a f-b e)+a c^2 \log ^2(f)}{c^2 \log ^2(f)+f^2}} \exp \left (-\frac{1}{4} i \left (\frac{b^2 \log ^2(f)}{f+i c \log (f)}+\frac{e^2}{f-i c \log (f)}\right )\right ) \left (\sqrt{f-i c \log (f)} (f+i c \log (f)) (\sin (d)-i \cos (d)) e^{\frac{i b^2 f \log ^2(f)}{2 \left (c^2 \log ^2(f)+f^2\right )}} f^{\frac{b e}{2 f+2 i c \log (f)}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-i \log (f) (b+2 c x)+e+2 f x)}{2 \sqrt{f-i c \log (f)}}\right )-(f-i c \log (f)) \sqrt{f+i c \log (f)} (\cos (d)-i \sin (d)) e^{\frac{i e^2 f}{2 \left (c^2 \log ^2(f)+f^2\right )}} f^{\frac{b e}{2 f-2 i c \log (f)}} \text{Erfi}\left (\frac{(-1)^{3/4} (i \log (f) (b+2 c x)+e+2 f x)}{2 \sqrt{f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.1, size = 214, normalized size = 1. \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{4}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-2\,i\ln \left ( f \right ) be+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{b\ln \left ( f \right ) -ie}{2}{\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{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+2\,i\ln \left ( f \right ) be-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{ie+b\ln \left ( f \right ) }{2}{\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 = 0.560543, size = 938, normalized size = 4.51 \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 \, f^{2} x +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + e f +{\left (i \, c e - i \, b f\right )} \log \left (f\right )\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{{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + i \, e^{2} f - 4 i \, d f^{2} -{\left (4 i \, c^{2} d - 2 i \, b c e + i \, b^{2} f\right )} \log \left (f\right )^{2} -{\left (c e^{2} - 2 \, b e f + 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 \, f^{2} x +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2} + e f +{\left (-i \, c e + i \, b f\right )} \log \left (f\right )\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{{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - i \, e^{2} f + 4 i \, d f^{2} -{\left (-4 i \, c^{2} d + 2 i \, b c e - i \, b^{2} f\right )} \log \left (f\right )^{2} -{\left (c e^{2} - 2 \, b e f + 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 + b x + 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} + b x + 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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