Optimal. Leaf size=126 \[ -\frac {3 \text {erfc}(b x)^2}{16 b^4}-\frac {x^3 e^{-b^2 x^2} \text {erfc}(b x)}{2 \sqrt {\pi } b}+\frac {x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^4}-\frac {3 x e^{-b^2 x^2} \text {erfc}(b x)}{4 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {erfc}(b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6365, 6386, 6374, 30, 2209, 2212} \[ -\frac {x^3 e^{-b^2 x^2} \text {Erfc}(b x)}{2 \sqrt {\pi } b}-\frac {3 x e^{-b^2 x^2} \text {Erfc}(b x)}{4 \sqrt {\pi } b^3}-\frac {3 \text {Erfc}(b x)^2}{16 b^4}+\frac {x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^4}+\frac {1}{4} x^4 \text {Erfc}(b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2209
Rule 2212
Rule 6365
Rule 6374
Rule 6386
Rubi steps
\begin {align*} \int x^3 \text {erfc}(b x)^2 \, dx &=\frac {1}{4} x^4 \text {erfc}(b x)^2+\frac {b \int e^{-b^2 x^2} x^4 \text {erfc}(b x) \, dx}{\sqrt {\pi }}\\ &=-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)^2-\frac {\int e^{-2 b^2 x^2} x^3 \, dx}{\pi }+\frac {3 \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx}{2 b \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)^2-\frac {\int e^{-2 b^2 x^2} x \, dx}{2 b^2 \pi }-\frac {3 \int e^{-2 b^2 x^2} x \, dx}{2 b^2 \pi }+\frac {3 \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{4 b^3 \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)^2-\frac {3 \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{8 b^4}\\ &=\frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erfc}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erfc}(b x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.43, size = 149, normalized size = 1.18 \[ \frac {1}{8} \left (\left (\frac {3}{b^4}-4 x^4\right ) \text {erf}(b x)+\frac {e^{-2 b^2 x^2} \left (4 \sqrt {\pi } b x e^{b^2 x^2} \left (2 b^2 x^2+3\right ) \text {erf}(b x)-3 \pi e^{2 b^2 x^2} \text {erf}(b x)^2+4 b^2 x^2+8\right )}{2 \pi b^4}-\frac {2 x e^{-b^2 x^2} \left (2 b^2 x^2+3\right )}{\sqrt {\pi } b^3}+2 x^4 \text {erf}(b x)^2+2 x^4\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 124, normalized size = 0.98 \[ \frac {4 \, \pi b^{4} x^{4} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x - {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} + 2 \, {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right ) + 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{3} \mathrm {erfc}\left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.24, size = 102, normalized size = 0.81 \[ \frac {x^4\,{\mathrm {erfc}\left (b\,x\right )}^2}{4}-\frac {\frac {3\,\pi \,{\mathrm {erfc}\left (b\,x\right )}^2}{16}-\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}-\frac {b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{2}+\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{4}}{b^4\,\pi } \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.86, size = 121, normalized size = 0.96 \[ \begin {cases} \frac {x^{4} \operatorname {erfc}^{2}{\left (b x \right )}}{4} - \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \pi b^{2}} - \frac {3 x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erfc}^{2}{\left (b x \right )}}{16 b^{4}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________