Integrand size = 18, antiderivative size = 144 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=-\frac {b}{10 \sqrt {\pi } x^4}+\frac {2 b^3}{15 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{15 x^3}-\frac {4 b^4 e^{-b^2 x^2} \text {erfi}(b x)}{15 x}-\frac {8 b^7 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{15 \sqrt {\pi }}+\frac {8 b^5 \log (x)}{15 \sqrt {\pi }} \]
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Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6528, 6513, 29, 30} \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=-\frac {8 b^7 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{15 \sqrt {\pi }}+\frac {8 b^5 \log (x)}{15 \sqrt {\pi }}+\frac {2 b^3}{15 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{15 x^3}-\frac {4 b^4 e^{-b^2 x^2} \text {erfi}(b x)}{15 x}-\frac {b}{10 \sqrt {\pi } x^4} \]
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Rule 29
Rule 30
Rule 6513
Rule 6528
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {1}{5} \left (2 b^2\right ) \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx+\frac {(2 b) \int \frac {1}{x^5} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b}{10 \sqrt {\pi } x^4}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{15 x^3}+\frac {1}{15} \left (4 b^4\right ) \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2} \, dx-\frac {\left (4 b^3\right ) \int \frac {1}{x^3} \, dx}{15 \sqrt {\pi }} \\ & = -\frac {b}{10 \sqrt {\pi } x^4}+\frac {2 b^3}{15 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{15 x^3}-\frac {4 b^4 e^{-b^2 x^2} \text {erfi}(b x)}{15 x}-\frac {1}{15} \left (8 b^6\right ) \int e^{-b^2 x^2} \text {erfi}(b x) \, dx+\frac {\left (8 b^5\right ) \int \frac {1}{x} \, dx}{15 \sqrt {\pi }} \\ & = -\frac {b}{10 \sqrt {\pi } x^4}+\frac {2 b^3}{15 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{15 x^3}-\frac {4 b^4 e^{-b^2 x^2} \text {erfi}(b x)}{15 x}-\frac {8 b^7 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{15 \sqrt {\pi }}+\frac {8 b^5 \log (x)}{15 \sqrt {\pi }} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=-\frac {b G_{2,3}^{2,1}\left (b^2 x^2|\begin {array}{c} 0,3 \\ 0,2,-\frac {1}{2} \\\end {array}\right )}{2 x^4} \]
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\[\int \frac {\operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{6}}d x\]
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\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}} \,d x } \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^6} \,d x \]
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