Optimal. Leaf size=105 \[ \frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6393, 6378, 29, 30} \[ \frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {2 b^2 e^{-b^2 x^2} \text {Erfi}(b x)}{3 x}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{3 x^3}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }}-\frac {b}{3 \sqrt {\pi } x^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 30
Rule 6378
Rule 6393
Rubi steps
\begin {align*} \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx &=-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx+\frac {(2 b) \int \frac {1}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{-b^2 x^2} \text {erfi}(b x) \, dx-\frac {\left (4 b^3\right ) \int \frac {1}{x} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}+\frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 29, normalized size = 0.28 \[ -\frac {b G_{2,3}^{2,1}\left (b^2 x^2|\begin {array}{c} 0,2 \\ 0,1,-\frac {1}{2} \\\end {array}\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{4}}\, dx \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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