Optimal. Leaf size=340 \[ \frac{3}{16} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{1}{4} i \left (4 d+\frac{(b \log (f)+i e)^2}{f}\right )} \text{Erf}\left (\frac{\sqrt [4]{-1} (b \log (f)+i e+2 i f x)}{2 \sqrt{f}}\right )+\left (\frac{1}{16}-\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{\frac{i (b \log (f)+3 i e)^2}{12 f}+3 i d} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (b \log (f)+3 i e+6 i f x)}{\sqrt{6} \sqrt{f}}\right )-\frac{3}{16} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{i (e+i b \log (f))^2}{4 f}-i d} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-b \log (f)+i e+2 i f x)}{2 \sqrt{f}}\right )-\left (\frac{1}{16}-\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{\frac{i (3 e+i b \log (f))^2}{12 f}-3 i d} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (-b \log (f)+3 i e+6 i f x)}{\sqrt{6} \sqrt{f}}\right ) \]
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Rubi [A] time = 0.601827, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4472, 2287, 2234, 2204, 2205} \[ \frac{3}{16} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{1}{4} i \left (4 d+\frac{(b \log (f)+i e)^2}{f}\right )} \text{Erf}\left (\frac{\sqrt [4]{-1} (b \log (f)+i e+2 i f x)}{2 \sqrt{f}}\right )+\left (\frac{1}{16}-\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{\frac{i (b \log (f)+3 i e)^2}{12 f}+3 i d} \text{Erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (b \log (f)+3 i e+6 i f x)}{\sqrt{6} \sqrt{f}}\right )-\frac{3}{16} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{i (e+i b \log (f))^2}{4 f}-i d} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-b \log (f)+i e+2 i f x)}{2 \sqrt{f}}\right )-\left (\frac{1}{16}-\frac{i}{16}\right ) \sqrt{\frac{\pi }{6}} f^{a-\frac{1}{2}} e^{\frac{i (3 e+i b \log (f))^2}{12 f}-3 i d} \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (-b \log (f)+3 i e+6 i f x)}{\sqrt{6} \sqrt{f}}\right ) \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2287
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int f^{a+b x} \sin ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac{1}{8} i e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x}+\frac{3}{8} i \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x}-\frac{3}{8} i \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x}+\frac{1}{8} i \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x}\right ) \, dx\\ &=-\left (\frac{1}{8} i \int e^{-3 i \left (d+e x+f x^2\right )} f^{a+b x} \, dx\right )+\frac{1}{8} i \int \exp \left (6 i d+6 i e x+6 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx+\frac{3}{8} i \int \exp \left (2 i d+2 i e x+2 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx-\frac{3}{8} i \int \exp \left (4 i d+4 i e x+4 i f x^2-3 i \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx\\ &=-\left (\frac{1}{8} i \int \exp \left (-3 i d-3 i f x^2+a \log (f)-x (3 i e-b \log (f))\right ) \, dx\right )+\frac{1}{8} i \int \exp \left (3 i d+3 i f x^2+a \log (f)+x (3 i e+b \log (f))\right ) \, dx+\frac{3}{8} i \int \exp \left (-i d-i f x^2+a \log (f)-x (i e-b \log (f))\right ) \, dx-\frac{3}{8} i \int \exp \left (i d+i f x^2+a \log (f)+x (i e+b \log (f))\right ) \, dx\\ &=-\left (\frac{1}{8} \left (i \exp \left (-3 i d+a \log (f)-\frac{i (-3 i e+b \log (f))^2}{12 f}\right )\right ) \int e^{\frac{i (-3 i e-6 i f x+b \log (f))^2}{12 f}} \, dx\right )+\frac{1}{8} \left (3 i e^{-i d+\frac{i (e+i b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac{i (-i e-2 i f x+b \log (f))^2}{4 f}} \, dx-\frac{1}{8} \left (3 i e^{\frac{1}{4} i \left (4 d+\frac{(i e+b \log (f))^2}{f}\right )} f^a\right ) \int e^{-\frac{i (i e+2 i f x+b \log (f))^2}{4 f}} \, dx+\frac{1}{8} \left (i e^{3 i d+\frac{i (3 i e+b \log (f))^2}{12 f}} f^a\right ) \int e^{-\frac{i (3 i e+6 i f x+b \log (f))^2}{12 f}} \, dx\\ &=\frac{3}{16} (-1)^{3/4} e^{\frac{1}{4} i \left (4 d+\frac{(i e+b \log (f))^2}{f}\right )} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erf}\left (\frac{\sqrt [4]{-1} (i e+2 i f x+b \log (f))}{2 \sqrt{f}}\right )+\left (\frac{1}{16}-\frac{i}{16}\right ) e^{3 i d+\frac{i (3 i e+b \log (f))^2}{12 f}} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{6}} \text{erf}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (3 i e+6 i f x+b \log (f))}{\sqrt{6} \sqrt{f}}\right )-\frac{3}{16} (-1)^{3/4} e^{-i d+\frac{i (e+i b \log (f))^2}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt [4]{-1} (i e+2 i f x-b \log (f))}{2 \sqrt{f}}\right )-\left (\frac{1}{16}-\frac{i}{16}\right ) \exp \left (-\frac{1}{12} i \left (36 d+\frac{(3 i e-b \log (f))^2}{f}\right )\right ) f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{6}} \text{erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (3 i e+6 i f x-b \log (f))}{\sqrt{6} \sqrt{f}}\right )\\ \end{align*}
Mathematica [A] time = 1.5456, size = 323, normalized size = 0.95 \[ \frac{1}{48} (-1)^{3/4} \sqrt{\pi } f^{a-\frac{b e+f}{2 f}} e^{-\frac{i \left (b^2 \log ^2(f)+3 e^2\right )}{4 f}} \left (9 i (\cos (d)+i \sin (d)) e^{\frac{i \left (b^2 \log ^2(f)+e^2\right )}{2 f}} \text{Erfi}\left (\frac{\sqrt [4]{-1} (-i b \log (f)+e+2 f x)}{2 \sqrt{f}}\right )+e^{\frac{i e^2}{f}} \left (\sqrt{3} (\cos (3 d)-i \sin (3 d)) e^{\frac{i \left (b^2 \log ^2(f)+3 e^2\right )}{6 f}} \text{Erfi}\left (\frac{(-1)^{3/4} (i b \log (f)+3 e+6 f x)}{2 \sqrt{3} \sqrt{f}}\right )-9 (\cos (d)-i \sin (d)) \text{Erfi}\left (\frac{(-1)^{3/4} (i b \log (f)+e+2 f x)}{2 \sqrt{f}}\right )\right )+\sqrt{3} e^{\frac{i b^2 \log ^2(f)}{3 f}} (\sin (3 d)-i \cos (3 d)) \text{Erfi}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) (-i b \log (f)+3 e+6 f x)}{\sqrt{6} \sqrt{f}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.486, size = 311, normalized size = 0.9 \begin{align*}{-{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{{\frac{{\frac{i}{12}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+6\,i\ln \left ( f \right ) be-9\,{e}^{2}+36\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{-3\,if}x+{\frac{3\,ie+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-3\,if}}}} \right ){\frac{1}{\sqrt{-3\,if}}}}+{{\frac{i}{48}}\sqrt{3}{f}^{a}\sqrt{\pi }{{\rm e}^{{\frac{-{\frac{i}{12}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-6\,i\ln \left ( f \right ) be-9\,{e}^{2}+36\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{3}\sqrt{if}x+{\frac{ \left ( b\ln \left ( f \right ) -3\,ie \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{if}}}} \right ){\frac{1}{\sqrt{if}}}}-{{\frac{3\,i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{{\frac{-{\frac{i}{4}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-2\,i\ln \left ( f \right ) be-{e}^{2}+4\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{if}x+{\frac{b\ln \left ( f \right ) -ie}{2}{\frac{1}{\sqrt{if}}}} \right ){\frac{1}{\sqrt{if}}}}+{{\frac{3\,i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{{\frac{{\frac{i}{4}} \left ( \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+2\,i\ln \left ( f \right ) be-{e}^{2}+4\,df \right ) }{f}}}}{\it Erf} \left ( -\sqrt{-if}x+{\frac{ie+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-if}}}} \right ){\frac{1}{\sqrt{-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 = 0.555856, size = 1773, normalized size = 5.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.55873, size = 1030, normalized size = 3.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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