Optimal. Leaf size=91 \[ \frac {i e^{i c} \sqrt {\pi } \text {Erfc}(b x)^2}{8 b}+\frac {i e^{-i c} \sqrt {\pi } \text {Erfi}(b x)}{4 b}-\frac {i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }} \]
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Rubi [A]
time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6540, 6512,
2235, 6511, 6509, 30} \begin {gather*} -\frac {i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {i \sqrt {\pi } e^{i c} \text {Erfc}(b x)^2}{8 b}+\frac {i \sqrt {\pi } e^{-i c} \text {Erfi}(b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2235
Rule 6509
Rule 6511
Rule 6512
Rule 6540
Rubi steps
\begin {align*} \int \text {erfc}(b x) \sin \left (c+i b^2 x^2\right ) \, dx &=-\left (\frac {1}{2} i \int e^{i c-b^2 x^2} \text {erfc}(b x) \, dx\right )+\frac {1}{2} i \int e^{-i c+b^2 x^2} \text {erfc}(b x) \, dx\\ &=\frac {1}{2} i \int e^{-i c+b^2 x^2} \, dx-\frac {1}{2} i \int e^{-i c+b^2 x^2} \text {erf}(b x) \, dx+\frac {\left (i e^{i c} \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x))}{4 b}\\ &=\frac {i e^{i c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {i e^{-i c} \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {i b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 94, normalized size = 1.03 \begin {gather*} \frac {(i \cos (c)+\sin (c)) \left (-4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )+\pi \left (2 \text {Erfi}(b x)-2 \text {Erf}(b x) (\cos (2 c)+i \sin (2 c))+\text {Erf}(b x)^2 (\cos (2 c)+i \sin (2 c))\right )\right )}{8 b \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \mathrm {erfc}\left (b x \right ) \sin \left (i b^{2} x^{2}+c \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin {\left (i b^{2} x^{2} + c \right )} \operatorname {erfc}{\left (b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sin \left (b^2\,x^2\,1{}\mathrm {i}+c\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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