Optimal. Leaf size=171 \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{c \log (f)}-2 i d} \text{Erfi}\left (\frac{-c x \log (f)+i e}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{e^2}{c \log (f)}+2 i d} \text{Erfi}\left (\frac{c x \log (f)+i e}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.217943, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4472, 2204, 2287, 2234} \[ \frac{\sqrt{\pi } f^a e^{\frac{e^2}{c \log (f)}-2 i d} \text{Erfi}\left (\frac{-c x \log (f)+i e}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}-\frac{\sqrt{\pi } f^a e^{\frac{e^2}{c \log (f)}+2 i d} \text{Erfi}\left (\frac{c x \log (f)+i e}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}+\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2204
Rule 2287
Rule 2234
Rubi steps
\begin{align*} \int f^{a+c x^2} \sin ^2(d+e x) \, dx &=\int \left (\frac{1}{2} f^{a+c x^2}-\frac{1}{4} e^{-2 i d-2 i e x} f^{a+c x^2}-\frac{1}{4} e^{2 i d+2 i e x} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{4} \int e^{-2 i d-2 i e x} f^{a+c x^2} \, dx\right )-\frac{1}{4} \int e^{2 i d+2 i e x} f^{a+c x^2} \, dx+\frac{1}{2} \int f^{a+c x^2} \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{1}{4} \int e^{-2 i d-2 i e x+a \log (f)+c x^2 \log (f)} \, dx-\frac{1}{4} \int e^{2 i d+2 i e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}-\frac{1}{4} \left (e^{-2 i d+\frac{e^2}{c \log (f)}} f^a\right ) \int e^{\frac{(-2 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx-\frac{1}{4} \left (e^{2 i d+\frac{e^2}{c \log (f)}} f^a\right ) \int e^{\frac{(2 i e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}}+\frac{e^{-2 i d+\frac{e^2}{c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e-c x \log (f)}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}-\frac{e^{2 i d+\frac{e^2}{c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+c x \log (f)}{\sqrt{c} \sqrt{\log (f)}}\right )}{8 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.250202, size = 132, normalized size = 0.77 \[ \frac{\sqrt{\pi } f^a \left (2 \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )-e^{\frac{e^2}{c \log (f)}} \left ((\cos (2 d)-i \sin (2 d)) \text{Erfi}\left (\frac{c x \log (f)-i e}{\sqrt{c} \sqrt{\log (f)}}\right )+(\cos (2 d)+i \sin (2 d)) \text{Erfi}\left (\frac{c x \log (f)+i e}{\sqrt{c} \sqrt{\log (f)}}\right )\right )\right )}{8 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 145, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{2\,id\ln \left ( f \right ) c-{e}^{2}}{c\ln \left ( f \right ) }}}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x+{ie{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{{f}^{a}\sqrt{\pi }}{8}{{\rm e}^{{\frac{2\,id\ln \left ( f \right ) c+{e}^{2}}{c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{ie{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{\frac{{f}^{a}\sqrt{\pi }}{4}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x \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 = 0.497576, size = 452, normalized size = 2.64 \begin{align*} -\frac{2 \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x\right ) - \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (c x \log \left (f\right ) + i \, e\right )} \sqrt{-c \log \left (f\right )}}{c \log \left (f\right )}\right ) e^{\left (\frac{a c \log \left (f\right )^{2} + 2 i \, c d \log \left (f\right ) + e^{2}}{c \log \left (f\right )}\right )} - \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (c x \log \left (f\right ) - i \, e\right )} \sqrt{-c \log \left (f\right )}}{c \log \left (f\right )}\right ) e^{\left (\frac{a c \log \left (f\right )^{2} - 2 i \, c d \log \left (f\right ) + e^{2}}{c \log \left (f\right )}\right )}}{8 \, 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 ^{2}{\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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