Optimal. Leaf size=71 \[ -\frac {1}{3} b^4 \text {erf}(b x)+\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)}{4 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 71, 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 = {6362, 2214, 2205} \[ -\frac {1}{3} b^4 \text {Erf}(b x)-\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}+\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {\text {Erfc}(b x)}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2205
Rule 2214
Rule 6362
Rubi steps
\begin {align*} \int \frac {\text {erfc}(b x)}{x^5} \, dx &=-\frac {\text {erfc}(b x)}{4 x^4}-\frac {b \int \frac {e^{-b^2 x^2}}{x^4} \, dx}{2 \sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {\text {erfc}(b x)}{4 x^4}+\frac {b^3 \int \frac {e^{-b^2 x^2}}{x^2} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)}{4 x^4}-\frac {\left (2 b^5\right ) \int e^{-b^2 x^2} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {1}{3} b^4 \text {erf}(b x)-\frac {\text {erfc}(b x)}{4 x^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 53, normalized size = 0.75 \[ \frac {1}{12} \left (-4 b^4 \text {erf}(b x)+\frac {2 e^{-b^2 x^2} \left (b-2 b^3 x^2\right )}{\sqrt {\pi } x^3}-\frac {3 \text {erfc}(b x)}{x^4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 58, normalized size = 0.82 \[ -\frac {3 \, \pi + 2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )}{12 \, \pi x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 69, normalized size = 0.97 \[ b^{4} \left (-\frac {\mathrm {erfc}\left (b x \right )}{4 b^{4} x^{4}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} x^{3}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3 b x}+\frac {2 \sqrt {\pi }\, \erf \left (b x \right )}{3}}{2 \sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.31, size = 35, normalized size = 0.49 \[ \frac {b^{4} {\left (x^{2}\right )}^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, b^{2} x^{2}\right )}{4 \, \sqrt {\pi } x^{3}} - \frac {\operatorname {erfc}\left (b x\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.19, size = 71, normalized size = 1.00 \[ -\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{4}+\frac {b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{3\,\sqrt {\pi }}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{6\,\sqrt {\pi }}}{x^4}-\frac {b^5\,\mathrm {erfi}\left (x\,\sqrt {-b^2}\right )}{3\,\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.14, size = 60, normalized size = 0.85 \[ \frac {b^{4} \operatorname {erfc}{\left (b x \right )}}{3} - \frac {b^{3} e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x} + \frac {b e^{- b^{2} x^{2}}}{6 \sqrt {\pi } x^{3}} - \frac {\operatorname {erfc}{\left (b x \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________