Optimal. Leaf size=107 \[ \frac{i \sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{4 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{4 \sqrt{c \log (f)+i f}} \]
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Rubi [A] time = 0.198138, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4472, 2287, 2205, 2204} \[ \frac{i \sqrt{\pi } e^{-i d} f^a \text{Erf}\left (x \sqrt{-c \log (f)+i f}\right )}{4 \sqrt{-c \log (f)+i f}}-\frac{i \sqrt{\pi } e^{i d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+i f}\right )}{4 \sqrt{c \log (f)+i f}} \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2287
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int f^{a+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+c x^2}-\frac{1}{2} i e^{i d+i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{2} i \int e^{-i d-i f x^2} f^{a+c x^2} \, dx-\frac{1}{2} i \int e^{i d+i f x^2} f^{a+c x^2} \, dx\\ &=\frac{1}{2} i \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx-\frac{1}{2} i \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx\\ &=\frac{i e^{-i d} f^a \sqrt{\pi } \text{erf}\left (x \sqrt{i f-c \log (f)}\right )}{4 \sqrt{i f-c \log (f)}}-\frac{i e^{i d} f^a \sqrt{\pi } \text{erfi}\left (x \sqrt{i f+c \log (f)}\right )}{4 \sqrt{i f+c \log (f)}}\\ \end{align*}
Mathematica [A] time = 0.462602, size = 170, normalized size = 1.59 \[ -\frac{\sqrt [4]{-1} \sqrt{\pi } f^a \left (\sqrt{f+i c \log (f)} \left (c \sin (d) \log (f) \text{Erf}\left (\frac{(1+i) x \sqrt{f+i c \log (f)}}{\sqrt{2}}\right )+\text{Erfi}\left ((-1)^{3/4} x \sqrt{f+i c \log (f)}\right ) (f \sin (d)+\cos (d) (c \log (f)+i f))\right )+\sqrt{f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) \text{Erfi}\left (\sqrt [4]{-1} x \sqrt{f-i c \log (f)}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.129, size = 84, normalized size = 0.8 \begin{align*}{-{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\rm e}^{id}}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -if}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -if}}}}+{{\frac{i}{4}}{f}^{a}\sqrt{\pi }{{\rm e}^{-id}}{\it Erf} \left ( x\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 [A] time = 0.499155, size = 301, normalized size = 2.81 \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 (\sqrt{-c \log \left (f\right ) - i \, f} x\right ) e^{\left (a \log \left (f\right ) + i \, d\right )} + \sqrt{\pi }{\left (-i \, c \log \left (f\right ) + f\right )} \sqrt{-c \log \left (f\right ) + i \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + i \, f} x\right ) e^{\left (a \log \left (f\right ) - i \, d\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}} \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} + 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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