Optimal. Leaf size=25 \[ -\frac {\text {Erfi}(b x)}{x}+\frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }} \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6498, 2241}
\begin {gather*} \frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {Erfi}(b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 6498
Rubi steps
\begin {align*} \int \frac {\text {erfi}(b x)}{x^2} \, dx &=-\frac {\text {erfi}(b x)}{x}+\frac {(2 b) \int \frac {e^{b^2 x^2}}{x} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erfi}(b x)}{x}+\frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\text {Erfi}(b x)}{x}+\frac {b \text {Ei}\left (b^2 x^2\right )}{\sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 31, normalized size = 1.24
method | result | size |
derivativedivides | \(b \left (-\frac {\erfi \left (b x \right )}{b x}-\frac {\expIntegral \left (1, -b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\) | \(31\) |
default | \(b \left (-\frac {\erfi \left (b x \right )}{b x}-\frac {\expIntegral \left (1, -b^{2} x^{2}\right )}{\sqrt {\pi }}\right )\) | \(31\) |
meijerg | \(\frac {b \left (-\frac {2 \sqrt {\pi }\, \erfi \left (b x \right )}{b x}-2 \ln \left (-b^{2} x^{2}\right )-2 \expIntegral \left (1, -b^{2} x^{2}\right )+4 \ln \left (x \right )+4 \ln \left (i b \right )\right )}{2 \sqrt {\pi }}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 23, normalized size = 0.92 \begin {gather*} \frac {b {\rm Ei}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 1.16 \begin {gather*} \frac {\sqrt {\pi } b x {\rm Ei}\left (b^{2} x^{2}\right ) - \pi \operatorname {erfi}\left (b x\right )}{\pi x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.56, size = 32, normalized size = 1.28 \begin {gather*} - \frac {b \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{\sqrt {\pi }} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{x} + \frac {i}{x} \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 [B]
time = 0.18, size = 23, normalized size = 0.92 \begin {gather*} \frac {b\,\mathrm {ei}\left (b^2\,x^2\right )}{\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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