Optimal. Leaf size=212 \[ \frac {i \sqrt {\pi } f^a \exp \left (-\frac {(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text {erf}\left (\frac {-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } f^a \exp \left (\frac {(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]
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Rubi [A] time = 0.57, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4472, 2287, 2234, 2205, 2204} \[ \frac {i \sqrt {\pi } f^a \exp \left (-\frac {(e+i b \log (f))^2}{-4 c \log (f)+4 i f}-i d\right ) \text {Erf}\left (\frac {-b \log (f)+2 x (-c \log (f)+i f)+i e}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } f^a \exp \left (\frac {(e-i b \log (f))^2}{4 c \log (f)+4 i f}+i d\right ) \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)+i e}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 4472
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \sin \left (d+e x+f x^2\right ) \, dx &=\int \left (\frac {1}{2} i e^{-i d-i e x-i f x^2} f^{a+b x+c x^2}-\frac {1}{2} i e^{i d+i e x+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{2} i \int e^{-i d-i e x-i f x^2} f^{a+b x+c x^2} \, dx-\frac {1}{2} i \int e^{i d+i e x+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{2} i \int \exp \left (-i d+a \log (f)-x (i e-b \log (f))-x^2 (i f-c \log (f))\right ) \, dx-\frac {1}{2} i \int \exp \left (i d+a \log (f)+x (i e+b \log (f))+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac {1}{2} \left (i \exp \left (-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(-i e+b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx-\frac {1}{2} \left (i \exp \left (i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(i e+b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=\frac {i \exp \left (-i d-\frac {(e+i b \log (f))^2}{4 i f-4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erf}\left (\frac {i e-b \log (f)+2 x (i f-c \log (f))}{2 \sqrt {i f-c \log (f)}}\right )}{4 \sqrt {i f-c \log (f)}}-\frac {i \exp \left (i d+\frac {(e-i b \log (f))^2}{4 i f+4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {i e+b \log (f)+2 x (i f+c \log (f))}{2 \sqrt {i f+c \log (f)}}\right )}{4 \sqrt {i f+c \log (f)}}\\ \end {align*}
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Mathematica [A] time = 2.20, size = 347, normalized size = 1.64 \[ -\frac {\sqrt [4]{-1} \sqrt {\pi } f^{\frac {f (a f-b e)+a c^2 \log ^2(f)}{c^2 \log ^2(f)+f^2}} \exp \left (-\frac {1}{4} i \left (\frac {b^2 \log ^2(f)}{f+i c \log (f)}+\frac {e^2}{f-i c \log (f)}\right )\right ) \left (\sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) e^{\frac {i b^2 f \log ^2(f)}{2 \left (c^2 \log ^2(f)+f^2\right )}} f^{\frac {b e}{2 f+2 i c \log (f)}} \text {erfi}\left (\frac {\sqrt [4]{-1} (-i \log (f) (b+2 c x)+e+2 f x)}{2 \sqrt {f-i c \log (f)}}\right )+(f-i c \log (f)) \sqrt {f+i c \log (f)} (\sin (d)+i \cos (d)) e^{\frac {i e^2 f}{2 \left (c^2 \log ^2(f)+f^2\right )}} f^{\frac {b e}{2 f-2 i c \log (f)}} \text {erfi}\left (\frac {(-1)^{3/4} (i \log (f) (b+2 c x)+e+2 f x)}{2 \sqrt {f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.04, size = 375, normalized size = 1.77 \[ \frac {\sqrt {\pi } {\left (i \, c \log \relax (f) + f\right )} \sqrt {-c \log \relax (f) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2} + e f + {\left (i \, c e - i \, b f\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f) - i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} + i \, e^{2} f - 4 i \, d f^{2} - {\left (4 i \, c^{2} d - 2 i \, b c e + i \, b^{2} f\right )} \log \relax (f)^{2} - {\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \relax (f)}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )} + \sqrt {\pi } {\left (-i \, c \log \relax (f) + f\right )} \sqrt {-c \log \relax (f) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2} + e f + {\left (-i \, c e + i \, b f\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f) + i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) e^{\left (-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} - i \, e^{2} f + 4 i \, d f^{2} - {\left (-4 i \, c^{2} d + 2 i \, b c e - i \, b^{2} f\right )} \log \relax (f)^{2} - {\left (c e^{2} - 2 \, b e f + 4 \, a f^{2}\right )} \log \relax (f)}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}} \]
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 \[ \int f^{c x^{2} + b x + a} \sin \left (f x^{2} + e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 216, normalized size = 1.02 \[ \frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d f -e^{2}+2 i \ln \relax (f ) b e -4 i d \ln \relax (f ) c +\ln \relax (f )^{2} b^{2}}{4 \left (i f +c \ln \relax (f )\right )}} \erf \left (-\sqrt {-i f -c \ln \relax (f )}\, x +\frac {i e +b \ln \relax (f )}{2 \sqrt {-i f -c \ln \relax (f )}}\right )}{4 \sqrt {-i f -c \ln \relax (f )}}-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d f -e^{2}-2 i \ln \relax (f ) b e +4 i d \ln \relax (f ) c +\ln \relax (f )^{2} b^{2}}{4 \left (-i f +c \ln \relax (f )\right )}} \erf \left (-x \sqrt {i f -c \ln \relax (f )}+\frac {-i e +b \ln \relax (f )}{2 \sqrt {i f -c \ln \relax (f )}}\right )}{4 \sqrt {i f -c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 1007, normalized size = 4.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{c\,x^2+b\,x+a}\,\sin \left (f\,x^2+e\,x+d\right ) \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \sin {\left (d + e x + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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