Optimal. Leaf size=151 \[ -\frac{i \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{i \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.207281, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4472, 2287, 2234, 2204} \[ -\frac{i \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{i \sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2287
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int f^{a+c x^2} \sin (d+e x) \, dx &=\int \left (\frac{1}{2} i e^{-i d-i e x} f^{a+c x^2}-\frac{1}{2} i e^{i d+i e x} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i d-i e x} f^{a+c x^2} \, dx-\frac{1}{2} i \int e^{i d+i e x} f^{a+c x^2} \, dx\\ &=\frac{1}{2} i \int e^{-i d-i e x+a \log (f)+c x^2 \log (f)} \, dx-\frac{1}{2} i \int e^{i d+i e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac{1}{2} \left (i 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 (i 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{i 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{i 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*}
Mathematica [A] time = 0.153063, size = 119, normalized size = 0.79 \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{4 c \log (f)}} \left (i (\cos (d)+i \sin (d)) \text{Erfi}\left (\frac{-2 c x \log (f)-i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+(\sin (d)+i \cos (d)) \text{Erfi}\left (\frac{2 c x \log (f)-i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.203, size = 123, normalized size = 0.8 \begin{align*}{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\rm e}^{{\frac{4\,id\ln \left ( f \right ) c+{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{{\frac{i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{4\,id\ln \left ( f \right ) c-{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{{\frac{i}{2}}e{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-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 [A] time = 0.496638, size = 413, normalized size = 2.74 \begin{align*} \frac{i \, \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 )} - i \, \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 )} \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}} \sin{\left (d + e x \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} \sin \left (e x + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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