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 = 0.21, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2204
Rule 2234
Rule 2287
Rule 4472
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*}
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Mathematica [A] time = 0.15, 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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fricas [A] time = 0.94, size = 144, normalized size = 0.95 \[ \frac {i \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 4 i \, c d \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )} - i \, \sqrt {\pi } \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - i \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 4 i \, c d \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c x^{2} + a} \sin \left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 123, normalized size = 0.81 \[ \frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 i d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}}+\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 i d \ln \relax (f ) c -e^{2}}{4 \ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {i e}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.36, size = 206, normalized size = 1.36 \[ -\frac {\sqrt {\pi } {\left (f^{a} {\left (i \, \cos \relax (d) + \sin \relax (d)\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} + \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (-i \, \cos \relax (d) + \sin \relax (d)\right )} \operatorname {erf}\left (x \overline {\sqrt {-c \log \relax (f)}} - \frac {1}{2} i \, e \overline {\frac {1}{\sqrt {-c \log \relax (f)}}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (i \, \cos \relax (d) - \sin \relax (d)\right )} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) + i \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )} + f^{a} {\left (-i \, \cos \relax (d) - \sin \relax (d)\right )} \operatorname {erf}\left (\frac {2 \, c x \log \relax (f) - i \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (\frac {e^{2}}{4 \, c \log \relax (f)}\right )}\right )} \sqrt {-c \log \relax (f)}}{8 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,x^2+a}\,\sin \left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + c x^{2}} \sin {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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