Optimal. Leaf size=81 \[ -\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}+\frac {b^5 \text {Ei}\left (-b^2 x^2\right )}{10 \sqrt {\pi }}+\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erf}(b x)}{5 x^5} \]
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Rubi [A] time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6361, 2214, 2210} \[ \frac {b^5 \text {Ei}\left (-b^2 x^2\right )}{10 \sqrt {\pi }}+\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {\text {Erf}(b x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rule 6361
Rubi steps
\begin {align*} \int \frac {\text {erf}(b x)}{x^6} \, dx &=-\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 \text {Ei}\left (-b^2 x^2\right )}{10 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.77 \[ \frac {b x e^{-b^2 x^2} \left (b^2 x^2-1\right )+b^5 x^5 \text {Ei}\left (-b^2 x^2\right )-2 \sqrt {\pi } \text {erf}(b x)}{10 \sqrt {\pi } x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 60, normalized size = 0.74 \[ -\frac {2 \, \pi \operatorname {erf}\left (b x\right ) - \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (-b^{2} x^{2}\right ) + {\left (b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 68, normalized size = 0.84 \[ -\frac {\operatorname {erf}\left (b x\right )}{5 \, x^{5}} + \frac {b^{10} x^{4} {\rm Ei}\left (-b^{2} x^{2}\right ) + b^{8} x^{2} e^{\left (-b^{2} x^{2}\right )} - b^{6} e^{\left (-b^{2} x^{2}\right )}}{10 \, \sqrt {\pi } b^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 0.88 \[ b^{5} \left (-\frac {\erf \left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{10 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{10 b^{2} x^{2}}-\frac {\Ei \left (1, b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 27, normalized size = 0.33 \[ -\frac {b^{5} \Gamma \left (-2, b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 65, normalized size = 0.80 \[ \frac {b^5\,\mathrm {ei}\left (-b^2\,x^2\right )}{10\,\sqrt {\pi }}-\frac {\mathrm {erf}\left (b\,x\right )}{5\,x^5}-\frac {\frac {b\,{\mathrm {e}}^{-b^2\,x^2}}{2}-\frac {b^3\,x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2}}{5\,x^4\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.39, size = 76, normalized size = 0.94 \[ - \frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{10 \sqrt {\pi }} + \frac {b^{3} e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} - \frac {b e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} + \frac {\operatorname {erfc}{\left (b x \right )}}{5 x^{5}} - \frac {1}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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