Optimal. Leaf size=354 \[ -\frac{3 i \sqrt{\pi } f^a e^{\frac{(e+i b \log (f))^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{i \sqrt{\pi } f^a e^{\frac{(3 e+i b \log (f))^2}{4 c \log (f)}-3 i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{3 i \sqrt{\pi } f^a e^{\frac{(e-i b \log (f))^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{i \sqrt{\pi } f^a e^{3 i d-\frac{(b \log (f)+3 i e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.489745, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4472, 2287, 2234, 2204} \[ -\frac{3 i \sqrt{\pi } f^a e^{\frac{(e+i b \log (f))^2}{4 c \log (f)}-i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{i \sqrt{\pi } f^a e^{\frac{(3 e+i b \log (f))^2}{4 c \log (f)}-3 i d} \text{Erfi}\left (\frac{-b \log (f)-2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{3 i \sqrt{\pi } f^a e^{\frac{(e-i b \log (f))^2}{4 c \log (f)}+i d} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{i \sqrt{\pi } f^a e^{3 i d-\frac{(b \log (f)+3 i e)^2}{4 c \log (f)}} \text{Erfi}\left (\frac{b \log (f)+2 c x \log (f)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 4472
Rule 2287
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x+c x^2} \sin ^3(d+e x) \, dx &=\int \left (\frac{3}{8} i e^{-i d-i e x} f^{a+b x+c x^2}-\frac{3}{8} i e^{i d+i e x} f^{a+b x+c x^2}-\frac{1}{8} i e^{-3 i d-3 i e x} f^{a+b x+c x^2}+\frac{1}{8} i e^{3 i d+3 i e x} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} i \int e^{-3 i d-3 i e x} f^{a+b x+c x^2} \, dx\right )+\frac{1}{8} i \int e^{3 i d+3 i e x} f^{a+b x+c x^2} \, dx+\frac{3}{8} i \int e^{-i d-i e x} f^{a+b x+c x^2} \, dx-\frac{3}{8} i \int e^{i d+i e x} f^{a+b x+c x^2} \, dx\\ &=-\left (\frac{1}{8} i \int \exp \left (-3 i d+a \log (f)+c x^2 \log (f)-x (3 i e-b \log (f))\right ) \, dx\right )+\frac{1}{8} i \int \exp \left (3 i d+a \log (f)+c x^2 \log (f)+x (3 i e+b \log (f))\right ) \, dx+\frac{3}{8} i \int \exp \left (-i d+a \log (f)+c x^2 \log (f)-x (i e-b \log (f))\right ) \, dx-\frac{3}{8} i \int \exp \left (i d+a \log (f)+c x^2 \log (f)+x (i e+b \log (f))\right ) \, dx\\ &=-\left (\frac{1}{8} \left (3 i e^{i d+\frac{(e-i b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\right )+\frac{1}{8} \left (3 i e^{-i d+\frac{(e+i b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(-i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx-\frac{1}{8} \left (i \exp \left (-3 i d+\frac{(3 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac{(-3 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac{1}{8} \left (i e^{3 i d-\frac{(3 i e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(3 i e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\\ &=-\frac{3 i e^{-i d+\frac{(e+i b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e-b \log (f)-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{i \exp \left (-3 i d+\frac{(3 e+i b \log (f))^2}{4 c \log (f)}\right ) f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e-b \log (f)-2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}-\frac{3 i e^{i d+\frac{(e-i b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{i e+b \log (f)+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}+\frac{i e^{3 i d-\frac{(3 i e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt{\pi } \text{erfi}\left (\frac{3 i e+b \log (f)+2 c x \log (f)}{2 \sqrt{c} \sqrt{\log (f)}}\right )}{16 \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 1.00392, size = 391, normalized size = 1.1 \[ \frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} e^{\frac{e (e-6 i b \log (f))}{4 c \log (f)}} \left (-\sin (3 d) e^{\frac{2 e^2}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+i \cos (3 d) e^{\frac{2 e^2}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)+3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )-\sin (3 d) e^{\frac{e (2 e+3 i b \log (f))}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)-3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )-i \cos (3 d) e^{\frac{e (2 e+3 i b \log (f))}{c \log (f)}} \text{Erfi}\left (\frac{\log (f) (b+2 c x)-3 i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+3 i e^{\frac{i b e}{c}} (\cos (d)+i \sin (d)) \text{Erfi}\left (\frac{-\log (f) (b+2 c x)-i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )+3 e^{\frac{2 i b e}{c}} (\sin (d)+i \cos (d)) \text{Erfi}\left (\frac{\log (f) (b+2 c x)-i e}{2 \sqrt{c} \sqrt{\log (f)}}\right )\right )}{16 \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.453, size = 338, normalized size = 1. \begin{align*}{-{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+6\,i\ln \left ( f \right ) be-12\,id\ln \left ( f \right ) c-9\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{3\,ie+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{{\frac{i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-6\,i\ln \left ( f \right ) be+12\,id\ln \left ( f \right ) c-9\,{e}^{2}}{4\,c\ln \left ( f \right ) }}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) -3\,ie}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}-{{\frac{3\,i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-2\,i\ln \left ( f \right ) be+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{b\ln \left ( f \right ) -ie}{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}}+{{\frac{3\,i}{16}}{f}^{a}\sqrt{\pi }{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+2\,i\ln \left ( f \right ) be-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{ie+b\ln \left ( f \right ) }{2}{\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 = 0.526348, size = 973, normalized size = 2.75 \begin{align*} \frac{3 i \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - e^{2} -{\left (4 i \, c d - 2 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} - 3 i \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - e^{2} -{\left (-4 i \, c d + 2 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} - i \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 3 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 9 \, e^{2} -{\left (12 i \, c d - 6 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )} + i \, \sqrt{\pi } \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 3 i \, e\right )} \sqrt{-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 9 \, e^{2} -{\left (-12 i \, c d + 6 i \, b e\right )} \log \left (f\right )}{4 \, c \log \left (f\right )}\right )}}{16 \, c \log \left (f\right )} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c x^{2} + b x + a} \sin \left (e x + d\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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