Optimal. Leaf size=105 \[ \frac{4 b^5 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{3 \sqrt{\pi }}+\frac{2 b^2 e^{-b^2 x^2} \text{Erfi}(b x)}{3 x}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{3 x^3}-\frac{4 b^3 \log (x)}{3 \sqrt{\pi }}-\frac{b}{3 \sqrt{\pi } x^2} \]
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Rubi [A] time = 0.0976992, antiderivative size = 105, normalized size of antiderivative = 1., 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 6393
Rule 6378
Rule 29
Rule 30
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*}
Mathematica [C] time = 0.016869, 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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Maple [F] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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