Optimal. Leaf size=193 \[ -\frac{i \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]
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Rubi [A] time = 0.384992, antiderivative size = 193, 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 = {4472, 2287, 2234, 2205, 2204} \[ -\frac{i \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text{Erf}\left (\frac{b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt{-c \log (f)+i f}}\right )}{4 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } f^a e^{i d-\frac{b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt{c \log (f)+i f}}\right )}{4 \sqrt{c \log (f)+i f}} \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2287
Rule 2234
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x+c x^2} \sin \left (d+f x^2\right ) \, dx &=\int \left (\frac{1}{2} i e^{-i d-i f x^2} f^{a+b x+c x^2}-\frac{1}{2} i e^{i d+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i d-i f x^2} f^{a+b x+c x^2} \, dx-\frac{1}{2} i \int e^{i d+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac{1}{2} i \int \exp \left (-i d+a \log (f)+b x \log (f)-x^2 (i f-c \log (f))\right ) \, dx-\frac{1}{2} i \int \exp \left (i d+a \log (f)+b x \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac{1}{2} \left (i e^{-i d+\frac{b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx-\frac{1}{2} \left (i e^{i d-\frac{b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=-\frac{i e^{-i d+\frac{b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{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{i e^{i d-\frac{b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{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 = 0.967958, size = 230, normalized size = 1.19 \[ -\frac{\sqrt [4]{-1} \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{-4 c \log (f)+4 i f}} \left (\sqrt{f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) e^{\frac{i b^2 f \log ^2(f)}{2 \left (c^2 \log ^2(f)+f^2\right )}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (2 f x-i \log (f) (b+2 c x))}{2 \sqrt{f-i c \log (f)}}\right )+\sqrt{f+i c \log (f)} (c \log (f)+i f) (\cos (d)-i \sin (d)) \text{Erfi}\left (\frac{(-1)^{3/4} (2 f x+i \log (f) (b+2 c x))}{2 \sqrt{f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.378, size = 180, normalized size = 0.9 \begin{align*}{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,id\ln \left ( f \right ) c+4\,df}{4\,if+4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -if}x+{\frac{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}}}}-{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+4\,id\ln \left ( f \right ) c+4\,df}{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 ) }{2}{\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{if-c\ln \left ( f \right ) }}}} \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 = 0.544457, size = 778, normalized size = 4.03 \begin{align*} \frac{\sqrt{\pi }{\left (i \, c \log \left (f\right ) + f\right )} \sqrt{-c \log \left (f\right ) - i \, f} \operatorname{erf}\left (\frac{{\left (2 \, f^{2} x - i \, b f \log \left (f\right ) +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\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 f^{2} \log \left (f\right ) -{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} + 4 i \, d f^{2} +{\left (4 i \, c^{2} d + i \, b^{2} f\right )} \log \left (f\right )^{2}}{4 \,{\left (c^{2} \log \left (f\right )^{2} + f^{2}\right )}}\right )} + \sqrt{\pi }{\left (-i \, c \log \left (f\right ) + f\right )} \sqrt{-c \log \left (f\right ) + i \, f} \operatorname{erf}\left (\frac{{\left (2 \, f^{2} x + i \, b f \log \left (f\right ) +{\left (2 \, c^{2} x + b c\right )} \log \left (f\right )^{2}\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 f^{2} \log \left (f\right ) -{\left (b^{2} c - 4 \, a c^{2}\right )} \log \left (f\right )^{3} - 4 i \, d f^{2} +{\left (-4 i \, c^{2} d - i \, b^{2} f\right )} \log \left (f\right )^{2}}{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}} \sin{\left (d + 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} \sin \left (f x^{2} + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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