Integrand size = 8, antiderivative size = 96 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)-\frac {\text {erf}(b x)}{6 x^6} \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2245, 2236} \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {4}{45} b^6 \text {erf}(b x)-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)}{6 x^6} \]
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Rule 2236
Rule 2245
Rule 6496
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erf}(b x)}{6 x^6}+\frac {b \int \frac {e^{-b^2 x^2}}{x^6} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {\text {erf}(b x)}{6 x^6}-\frac {\left (2 b^3\right ) \int \frac {e^{-b^2 x^2}}{x^4} \, dx}{15 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)}{6 x^6}+\frac {\left (4 b^5\right ) \int \frac {e^{-b^2 x^2}}{x^2} \, dx}{45 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {\text {erf}(b x)}{6 x^6}-\frac {\left (8 b^7\right ) \int e^{-b^2 x^2} \, dx}{45 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)-\frac {\text {erf}(b x)}{6 x^6} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.76 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=\frac {e^{-b^2 x^2} \left (-6 b x+4 b^3 x^3-8 b^5 x^5-e^{b^2 x^2} \sqrt {\pi } \left (15+8 b^6 x^6\right ) \text {erf}(b x)\right )}{90 \sqrt {\pi } x^6} \]
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Time = 0.62 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.73
method | result | size |
meijerg | \(\frac {b^{6} \left (-\frac {4 \left (\frac {2}{9} b^{4} x^{4}-\frac {1}{9} b^{2} x^{2}+\frac {1}{6}\right ) {\mathrm e}^{-b^{2} x^{2}}}{5 x^{5} b^{5}}-\frac {\left (8 b^{6} x^{6}+15\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{45 x^{6} b^{6}}\right )}{2 \sqrt {\pi }}\) | \(70\) |
parallelrisch | \(-\frac {8 \,\operatorname {erf}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }+8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}-4 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+6 \,{\mathrm e}^{-b^{2} x^{2}} b x +15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{90 \sqrt {\pi }\, x^{6}}\) | \(81\) |
parts | \(-\frac {\operatorname {erf}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 x^{5}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{3}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{3}\right )}{5}\right )}{3 \sqrt {\pi }}\) | \(82\) |
derivativedivides | \(b^{6} \left (-\frac {\operatorname {erf}\left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\) | \(87\) |
default | \(b^{6} \left (-\frac {\operatorname {erf}\left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\) | \(87\) |
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Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} + {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{90 \, \pi x^{6}} \]
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Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=- \frac {4 b^{6} \operatorname {erf}{\left (b x \right )}}{45} - \frac {4 b^{5} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x} + \frac {2 b^{3} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x^{3}} - \frac {b e^{- b^{2} x^{2}}}{15 \sqrt {\pi } x^{5}} - \frac {\operatorname {erf}{\left (b x \right )}}{6 x^{6}} \]
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Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {b^{6} {\left (x^{2}\right )}^{\frac {5}{2}} \Gamma \left (-\frac {5}{2}, b^{2} x^{2}\right )}{6 \, \sqrt {\pi } x^{5}} - \frac {\operatorname {erf}\left (b x\right )}{6 \, x^{6}} \]
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\[ \int \frac {\text {erf}(b x)}{x^7} \, dx=\int { \frac {\operatorname {erf}\left (b x\right )}{x^{7}} \,d x } \]
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Time = 5.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {\text {erf}(b x)}{x^7} \, dx=-\frac {\mathrm {erf}\left (b\,x\right )}{6\,x^6}-\frac {3\,b\,{\mathrm {e}}^{-b^2\,x^2}-2\,b^3\,x^2\,{\mathrm {e}}^{-b^2\,x^2}+4\,b^5\,x^4\,{\mathrm {e}}^{-b^2\,x^2}+4\,b^5\,\sqrt {\pi }\,\sqrt {b^2}\,{\left (x^2\right )}^{5/2}-4\,b^5\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b^2}\,\sqrt {x^2}\right )\,\sqrt {b^2}\,{\left (x^2\right )}^{5/2}}{45\,x^5\,\sqrt {\pi }} \]
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