Optimal. Leaf size=107 \[ \frac {i \sqrt {\pi } e^{-i d} f^a \text {erf}\left (x \sqrt {-c \log (f)+i f}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } e^{i d} f^a \text {erfi}\left (x \sqrt {c \log (f)+i f}\right )}{4 \sqrt {c \log (f)+i f}} \]
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Rubi [A] time = 0.20, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4472, 2287, 2205, 2204} \[ \frac {i \sqrt {\pi } e^{-i d} f^a \text {Erf}\left (x \sqrt {-c \log (f)+i f}\right )}{4 \sqrt {-c \log (f)+i f}}-\frac {i \sqrt {\pi } e^{i d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+i f}\right )}{4 \sqrt {c \log (f)+i f}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2287
Rule 4472
Rubi steps
\begin {align*} \int f^{a+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+c x^2}-\frac {1}{2} i e^{i d+i f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{2} i \int e^{-i d-i f x^2} f^{a+c x^2} \, dx-\frac {1}{2} i \int e^{i d+i f x^2} f^{a+c x^2} \, dx\\ &=\frac {1}{2} i \int e^{-i d+a \log (f)-x^2 (i f-c \log (f))} \, dx-\frac {1}{2} i \int e^{i d+a \log (f)+x^2 (i f+c \log (f))} \, dx\\ &=\frac {i e^{-i d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {i f-c \log (f)}\right )}{4 \sqrt {i f-c \log (f)}}-\frac {i e^{i d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {i f+c \log (f)}\right )}{4 \sqrt {i f+c \log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 170, normalized size = 1.59 \[ -\frac {\sqrt [4]{-1} \sqrt {\pi } f^a \left (\sqrt {f+i c \log (f)} \left (c \sin (d) \log (f) \text {erf}\left (\frac {(1+i) x \sqrt {f+i c \log (f)}}{\sqrt {2}}\right )+\text {erfi}\left ((-1)^{3/4} x \sqrt {f+i c \log (f)}\right ) (f \sin (d)+\cos (d) (c \log (f)+i f))\right )+\sqrt {f-i c \log (f)} (f+i c \log (f)) (\cos (d)+i \sin (d)) \text {erfi}\left (\sqrt [4]{-1} x \sqrt {f-i c \log (f)}\right )\right )}{4 \left (c^2 \log ^2(f)+f^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 107, normalized size = 1.00 \[ \frac {\sqrt {\pi } {\left (i \, c \log \relax (f) + f\right )} \sqrt {-c \log \relax (f) - i \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - i \, f} x\right ) e^{\left (a \log \relax (f) + i \, d\right )} + \sqrt {\pi } {\left (-i \, c \log \relax (f) + f\right )} \sqrt {-c \log \relax (f) + i \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + i \, f} x\right ) e^{\left (a \log \relax (f) - i \, d\right )}}{4 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}} \]
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 \[ \int f^{c x^{2} + a} \sin \left (f x^{2} + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 84, normalized size = 0.79 \[ -\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{i d} \erf \left (\sqrt {-i f -c \ln \relax (f )}\, x \right )}{4 \sqrt {-i f -c \ln \relax (f )}}+\frac {i \sqrt {\pi }\, f^{a} {\mathrm e}^{-i d} \erf \left (x \sqrt {i f -c \ln \relax (f )}\right )}{4 \sqrt {i f -c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 209, normalized size = 1.95 \[ \frac {\sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 2 \, f^{2}} {\left (f^{a} {\left (\cos \relax (d) - i \, \sin \relax (d)\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + i \, f} x\right ) + f^{a} {\left (\cos \relax (d) + i \, \sin \relax (d)\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - i \, f} x\right )\right )} \sqrt {c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + f^{2}}} - \sqrt {\pi } \sqrt {2 \, c^{2} \log \relax (f)^{2} + 2 \, f^{2}} {\left (f^{a} {\left (-i \, \cos \relax (d) - \sin \relax (d)\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + i \, f} x\right ) + f^{a} {\left (i \, \cos \relax (d) - \sin \relax (d)\right )} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - i \, f} x\right )\right )} \sqrt {-c \log \relax (f) + \sqrt {c^{2} \log \relax (f)^{2} + f^{2}}}}{8 \, {\left (c^{2} \log \relax (f)^{2} + f^{2}\right )}} \]
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 \[ \int f^{c\,x^2+a}\,\sin \left (f\,x^2+d\right ) \,d x \]
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 \[ \int f^{a + c x^{2}} \sin {\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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