Optimal. Leaf size=59 \[ -\frac {e^{c+b^2 x^2} \text {Erfi}(b x)}{x}+\frac {1}{2} b e^c \sqrt {\pi } \text {Erfi}(b x)^2+\frac {b e^c \text {Ei}\left (2 b^2 x^2\right )}{\sqrt {\pi }} \]
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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6528, 6510, 30,
2241} \begin {gather*} -\frac {e^{b^2 x^2+c} \text {Erfi}(b x)}{x}+\frac {b e^c \text {Ei}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b e^c \text {Erfi}(b x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2241
Rule 6510
Rule 6528
Rubi steps
\begin {align*} \int \frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x^2} \, dx &=-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx+\frac {(2 b) \int \frac {e^{c+2 b^2 x^2}}{x} \, dx}{\sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\frac {b e^c \text {Ei}\left (2 b^2 x^2\right )}{\sqrt {\pi }}+\left (b e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x))\\ &=-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{x}+\frac {1}{2} b e^c \sqrt {\pi } \text {erfi}(b x)^2+\frac {b e^c \text {Ei}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.95 \begin {gather*} \frac {1}{2} e^c \left (-\frac {2 e^{b^2 x^2} \text {Erfi}(b x)}{x}+b \sqrt {\pi } \text {Erfi}(b x)^2+\frac {2 b \text {Ei}\left (2 b^2 x^2\right )}{\sqrt {\pi }}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \erfi \left (b x \right )}{x^{2}}\, dx\]
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 [A]
time = 0.34, size = 55, normalized size = 0.93 \begin {gather*} -\frac {{\left (2 \, \pi \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b x \operatorname {erfi}\left (b x\right )^{2} + 2 \, b x {\rm Ei}\left (2 \, b^{2} x^{2}\right )\right )}\right )} e^{c}}{2 \, \pi x} \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*} e^{c} \int \frac {e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{x^{2}}\, 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 = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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