Optimal. Leaf size=193 \[ -\frac {i \sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text {erf}\left (\frac {b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } f^a e^{i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text {erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]
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Rubi [A] time = 0.38, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4472, 2287, 2234, 2205, 2204} \[ -\frac {i \sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+4 i f}-i d} \text {Erf}\left (\frac {b \log (f)-2 x (-c \log (f)+i f)}{2 \sqrt {-c \log (f)+i f}}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } f^a e^{i d-\frac {b^2 \log ^2(f)}{4 c \log (f)+4 i f}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+i f)}{2 \sqrt {c \log (f)+i f}}\right )}{4 \sqrt {c \log (f)+i f}} \]
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+f x^2\right ) \, dx &=\int \left (\frac {1}{2} i e^{-i d-i f x^2} f^{a+b x+c x^2}-\frac {1}{2} i e^{i d+i f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{2} i \int e^{-i d-i f x^2} f^{a+b x+c x^2} \, dx-\frac {1}{2} i \int e^{i d+i f x^2} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{2} i \int \exp \left (-i d+a \log (f)+b x \log (f)-x^2 (i f-c \log (f))\right ) \, dx-\frac {1}{2} i \int \exp \left (i d+a \log (f)+b x \log (f)+x^2 (i f+c \log (f))\right ) \, dx\\ &=\frac {1}{2} \left (i e^{-i d+\frac {b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-i f+c \log (f)))^2}{4 (-i f+c \log (f))}\right ) \, dx-\frac {1}{2} \left (i e^{i d-\frac {b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (i f+c \log (f)))^2}{4 (i f+c \log (f))}\right ) \, dx\\ &=-\frac {i e^{-i d+\frac {b^2 \log ^2(f)}{4 i f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {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 e^{i d-\frac {b^2 \log ^2(f)}{4 i f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {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 = 0.99, size = 230, normalized size = 1.19 \[ -\frac {\sqrt [4]{-1} \sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{-4 c \log (f)+4 i f}} \left (\sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) e^{\frac {i b^2 f \log ^2(f)}{2 \left (c^2 \log ^2(f)+f^2\right )}} \text {erfi}\left (\frac {\sqrt [4]{-1} (2 f x-i \log (f) (b+2 c x))}{2 \sqrt {f-i c \log (f)}}\right )+\sqrt {f+i c \log (f)} (c \log (f)+i f) (\cos (d)-i \sin (d)) \text {erfi}\left (\frac {(-1)^{3/4} (2 f x+i \log (f) (b+2 c x))}{2 \sqrt {f+i c \log (f)}}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 309, normalized size = 1.60 \[ \frac {\sqrt {\pi } {\left (i \, c \log \relax (f) + f\right )} \sqrt {-c \log \relax (f) - i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x - i \, b f \log \relax (f) + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2}\right )} \sqrt {-c \log \relax (f) - i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a f^{2} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} + 4 i \, d f^{2} + {\left (4 i \, c^{2} d + i \, b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )} + \sqrt {\pi } {\left (-i \, c \log \relax (f) + f\right )} \sqrt {-c \log \relax (f) + i \, f} \operatorname {erf}\left (\frac {{\left (2 \, f^{2} x + i \, b f \log \relax (f) + {\left (2 \, c^{2} x + b c\right )} \log \relax (f)^{2}\right )} \sqrt {-c \log \relax (f) + i \, f}}{2 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) e^{\left (\frac {4 \, a f^{2} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)^{3} - 4 i \, d f^{2} + {\left (-4 i \, c^{2} d - i \, b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}} \]
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 \[ \int f^{c x^{2} + b x + a} \sin \left (f x^{2} + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.63, size = 180, normalized size = 0.93 \[ \frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d f -4 i d \ln \relax (f ) c +\ln \relax (f )^{2} b^{2}}{4 \left (i f +c \ln \relax (f )\right )}} \erf \left (-\sqrt {-i f -c \ln \relax (f )}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-i f -c \ln \relax (f )}}\right )}{4 \sqrt {-i f -c \ln \relax (f )}}-\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d f +4 i d \ln \relax (f ) c +\ln \relax (f )^{2} b^{2}}{4 \left (-i f +c \ln \relax (f )\right )}} \erf \left (-x \sqrt {i f -c \ln \relax (f )}+\frac {\ln \relax (f ) b}{2 \sqrt {i f -c \ln \relax (f )}}\right )}{4 \sqrt {i f -c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 647, normalized size = 3.35 \[ -\frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 2 \, f^{2}} {\left ({\left (f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) - i \, f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) - i \, f\right )} x + b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f) + i \, f}}\right ) + {\left (f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) + i \, f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) + i \, f\right )} x + b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f) - i \, f}}\right )\right )} \sqrt {c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + f^{2}}} + \sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 2 \, f^{2}} {\left ({\left (i \, f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) + f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) - i \, f\right )} x + b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f) + i \, f}}\right ) + {\left (-i \, f^{a} \cos \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right ) + f^{a} \sin \left (\frac {4 \, d f^{2} + {\left (4 \, c^{2} d + b^{2} f\right )} \log \relax (f)^{2}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )\right )} \operatorname {erf}\left (\frac {2 \, {\left (c \log \relax (f) + i \, f\right )} x + b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f) - i \, f}}\right )\right )} \sqrt {-c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + f^{2}}}}{8 \, {\left (c^{2} e^{\left (\frac {b^{2} c \log \relax (f)^{3}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )} \log \relax (f)^{2} + f^{2} e^{\left (\frac {b^{2} c \log \relax (f)^{3}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}}\right )}\right )}} \]
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 \[ \int f^{c\,x^2+b\,x+a}\,\sin \left (f\,x^2+d\right ) \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \sin {\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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