Optimal. Leaf size=96 \[ \frac {4}{45} b^6 \text {erf}(b x)+\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erfc}(b x)}{6 x^6} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6362, 2214, 2205} \[ \frac {4}{45} b^6 \text {Erf}(b x)+\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}+\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {\text {Erfc}(b x)}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2214
Rule 6362
Rubi steps
\begin {align*} \int \frac {\text {erfc}(b x)}{x^7} \, dx &=-\frac {\text {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(b x)}{6 x^6}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 62, normalized size = 0.65 \[ \frac {1}{90} \left (8 b^6 \text {erf}(b x)+\frac {2 b e^{-b^2 x^2} \left (4 b^4 x^4-2 b^2 x^2+3\right )}{\sqrt {\pi } x^5}-\frac {15 \text {erfc}(b x)}{x^6}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 66, normalized size = 0.69 \[ -\frac {15 \, \pi - 2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{90 \, \pi x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 0.91 \[ b^{6} \left (-\frac {\mathrm {erfc}\left (b x \right )}{6 b^{6} x^{6}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \sqrt {\pi }\, \erf \left (b x \right )}{15}}{3 \sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 35, normalized size = 0.36 \[ \frac {b^{6} {\left (x^{2}\right )}^{\frac {5}{2}} \Gamma \left (-\frac {5}{2}, b^{2} x^{2}\right )}{6 \, \sqrt {\pi } x^{5}} - \frac {\operatorname {erfc}\left (b x\right )}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {erfc}\left (b\,x\right )}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.85, size = 87, normalized size = 0.91 \[ - \frac {4 b^{6} \operatorname {erfc}{\left (b x \right )}}{45} + \frac {4 b^{5} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x} - \frac {2 b^{3} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x^{3}} + \frac {b e^{- b^{2} x^{2}}}{15 \sqrt {\pi } x^{5}} - \frac {\operatorname {erfc}{\left (b x \right )}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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