Optimal. Leaf size=65 \[ -\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)^2+\frac {2 b^2 \text {Ei}\left (2 b^2 x^2\right )}{\pi }-\frac {\text {erfi}(b x)^2}{2 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6366, 6393, 6375, 30, 2210} \[ -\frac {2 b e^{b^2 x^2} \text {Erfi}(b x)}{\sqrt {\pi } x}+b^2 \text {Erfi}(b x)^2+\frac {2 b^2 \text {Ei}\left (2 b^2 x^2\right )}{\pi }-\frac {\text {Erfi}(b x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2210
Rule 6366
Rule 6375
Rule 6393
Rubi steps
\begin {align*} \int \frac {\text {erfi}(b x)^2}{x^3} \, dx &=-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {(2 b) \int \frac {e^{b^2 x^2} \text {erfi}(b x)}{x^2} \, dx}{\sqrt {\pi }}\\ &=-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {\left (4 b^2\right ) \int \frac {e^{2 b^2 x^2}}{x} \, dx}{\pi }+\frac {\left (4 b^3\right ) \int e^{b^2 x^2} \text {erfi}(b x) \, dx}{\sqrt {\pi }}\\ &=-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {2 b^2 \text {Ei}\left (2 b^2 x^2\right )}{\pi }+\left (2 b^2\right ) \operatorname {Subst}(\int x \, dx,x,\text {erfi}(b x))\\ &=-\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)^2-\frac {\text {erfi}(b x)^2}{2 x^2}+\frac {2 b^2 \text {Ei}\left (2 b^2 x^2\right )}{\pi }\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.92 \[ -\frac {2 b e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } x}+\left (b^2-\frac {1}{2 x^2}\right ) \text {erfi}(b x)^2+\frac {2 b^2 \text {Ei}\left (2 b^2 x^2\right )}{\pi } \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 64, normalized size = 0.98 \[ \frac {4 \, b^{2} x^{2} {\rm Ei}\left (2 \, b^{2} x^{2}\right ) - 4 \, \sqrt {\pi } b x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )^{2}}{2 \, \pi x^{2}} \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (b x \right )^{2}}{x^{3}}\, dx \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right )^{2}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {erfi}\left (b\,x\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {erfi}^{2}{\left (b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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