Integrand size = 8, antiderivative size = 81 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=-\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}+\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{5 x^5}+\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2245, 2241} \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=-\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}+\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }}+\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{5 x^5} \]
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Rule 2241
Rule 2245
Rule 6496
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erf}(b x)}{5 x^5}+\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x^5} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {\text {erf}(b x)}{5 x^5}-\frac {b^3 \int \frac {e^{-b^2 x^2}}{x^3} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}+\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{5 x^5}+\frac {b^5 \int \frac {e^{-b^2 x^2}}{x} \, dx}{5 \sqrt {\pi }} \\ & = -\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}+\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{5 x^5}+\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=\frac {b e^{-b^2 x^2} x \left (-1+b^2 x^2\right )-2 \sqrt {\pi } \text {erf}(b x)+b^5 x^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi } x^5} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.66 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78
method | result | size |
meijerg | \(\frac {b^{5} \left (-\frac {b^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{2}\right ], \left [2, 4, \frac {9}{2}\right ], -b^{2} x^{2}\right )}{21}-\frac {19}{50}+\frac {\gamma }{5}+\frac {2 \ln \left (x \right )}{5}+\frac {2 \ln \left (b \right )}{5}-\frac {1}{b^{4} x^{4}}+\frac {2}{3 b^{2} x^{2}}\right )}{2 \sqrt {\pi }}\) | \(63\) |
parts | \(-\frac {\operatorname {erf}\left (b x \right )}{5 x^{5}}+\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{4}}-\frac {b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{2} \operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{2}\right )}{5 \sqrt {\pi }}\) | \(66\) |
derivativedivides | \(b^{5} \left (-\frac {\operatorname {erf}\left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{10 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{10 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right )\) | \(71\) |
default | \(b^{5} \left (-\frac {\operatorname {erf}\left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{10 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{10 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right )\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=-\frac {2 \, \pi \operatorname {erf}\left (b x\right ) - \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (-b^{2} x^{2}\right ) + {\left (b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \]
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Time = 1.77 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=- \frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{10 \sqrt {\pi }} + \frac {b^{3} e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} - \frac {b e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} + \frac {\operatorname {erfc}{\left (b x \right )}}{5 x^{5}} - \frac {1}{5 x^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.33 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=-\frac {b^{5} \Gamma \left (-2, b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )}{5 \, x^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=-\frac {\operatorname {erf}\left (b x\right )}{5 \, x^{5}} + \frac {b^{10} x^{4} {\rm Ei}\left (-b^{2} x^{2}\right ) + b^{8} x^{2} e^{\left (-b^{2} x^{2}\right )} - b^{6} e^{\left (-b^{2} x^{2}\right )}}{10 \, \sqrt {\pi } b^{5} x^{4}} \]
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Time = 5.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80 \[ \int \frac {\text {erf}(b x)}{x^6} \, dx=\frac {b^5\,\mathrm {ei}\left (-b^2\,x^2\right )}{10\,\sqrt {\pi }}-\frac {\mathrm {erf}\left (b\,x\right )}{5\,x^5}-\frac {\frac {b\,{\mathrm {e}}^{-b^2\,x^2}}{2}-\frac {b^3\,x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2}}{5\,x^4\,\sqrt {\pi }} \]
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