Optimal. Leaf size=178 \[ -\frac {5 \text {erfc}(b x)^2}{16 b^6}-\frac {x^5 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } b}+\frac {x^4 e^{-2 b^2 x^2}}{6 \pi b^2}+\frac {11 e^{-2 b^2 x^2}}{12 \pi b^6}-\frac {5 x e^{-b^2 x^2} \text {erfc}(b x)}{4 \sqrt {\pi } b^5}+\frac {7 x^2 e^{-2 b^2 x^2}}{12 \pi b^4}-\frac {5 x^3 e^{-b^2 x^2} \text {erfc}(b x)}{6 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erfc}(b x)^2 \]
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Rubi [A] time = 0.28, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ -\frac {x^5 e^{-b^2 x^2} \text {Erfc}(b x)}{3 \sqrt {\pi } b}-\frac {5 x^3 e^{-b^2 x^2} \text {Erfc}(b x)}{6 \sqrt {\pi } b^3}-\frac {5 x e^{-b^2 x^2} \text {Erfc}(b x)}{4 \sqrt {\pi } b^5}-\frac {5 \text {Erfc}(b x)^2}{16 b^6}+\frac {x^4 e^{-2 b^2 x^2}}{6 \pi b^2}+\frac {7 x^2 e^{-2 b^2 x^2}}{12 \pi b^4}+\frac {11 e^{-2 b^2 x^2}}{12 \pi b^6}+\frac {1}{6} x^6 \text {Erfc}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2209
Rule 2212
Rule 6365
Rule 6374
Rule 6386
Rubi steps
\begin {align*} \int x^5 \text {erfc}(b x)^2 \, dx &=\frac {1}{6} x^6 \text {erfc}(b x)^2+\frac {(2 b) \int e^{-b^2 x^2} x^6 \text {erfc}(b x) \, dx}{3 \sqrt {\pi }}\\ &=-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {2 \int e^{-2 b^2 x^2} x^5 \, dx}{3 \pi }+\frac {5 \int e^{-b^2 x^2} x^4 \text {erfc}(b x) \, dx}{3 b \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {2 \int e^{-2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac {5 \int e^{-2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }+\frac {5 \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx}{2 b^3 \sqrt {\pi }}\\ &=\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {\int e^{-2 b^2 x^2} x \, dx}{3 b^4 \pi }-\frac {5 \int e^{-2 b^2 x^2} x \, dx}{6 b^4 \pi }-\frac {5 \int e^{-2 b^2 x^2} x \, dx}{2 b^4 \pi }+\frac {5 \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{4 b^5 \sqrt {\pi }}\\ &=\frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {5 \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{8 b^6}\\ &=\frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}-\frac {5 \text {erfc}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x)^2\\ \end {align*}
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Mathematica [A] time = 0.47, size = 173, normalized size = 0.97 \[ \frac {1}{24} \left (\left (\frac {15}{b^6}-8 x^6\right ) \text {erf}(b x)+\frac {e^{-2 b^2 x^2} \left (8 b^4 x^4-15 \pi e^{2 b^2 x^2} \text {erf}(b x)^2+28 b^2 x^2+4 \sqrt {\pi } b x e^{b^2 x^2} \left (4 b^4 x^4+10 b^2 x^2+15\right ) \text {erf}(b x)+44\right )}{2 \pi b^6}-\frac {2 x e^{-b^2 x^2} \left (4 b^4 x^4+10 b^2 x^2+15\right )}{\sqrt {\pi } b^5}+4 x^6 \text {erf}(b x)^2+4 x^6\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 149, normalized size = 0.84 \[ \frac {8 \, \pi b^{6} x^{6} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} - 4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x - {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} + 2 \, {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right ) + 4 \, {\left (2 \, b^{4} x^{4} + 7 \, b^{2} x^{2} + 11\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \]
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 \[ \int x^{5} \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{5} \mathrm {erfc}\left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 143, normalized size = 0.80 \[ \frac {x^6\,{\mathrm {erfc}\left (b\,x\right )}^2}{6}-\frac {\frac {5\,\pi \,{\mathrm {erfc}\left (b\,x\right )}^2}{16}-\frac {11\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}-\frac {7\,b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}-\frac {b^4\,x^4\,{\mathrm {e}}^{-2\,b^2\,x^2}}{6}+\frac {5\,b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{6}+\frac {b^5\,x^5\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{3}+\frac {5\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{4}}{b^6\,\pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.09, size = 172, normalized size = 0.97 \[ \begin {cases} \frac {x^{6} \operatorname {erfc}^{2}{\left (b x \right )}}{6} - \frac {x^{5} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{3 \sqrt {\pi } b} + \frac {x^{4} e^{- 2 b^{2} x^{2}}}{6 \pi b^{2}} - \frac {5 x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{6 \sqrt {\pi } b^{3}} + \frac {7 x^{2} e^{- 2 b^{2} x^{2}}}{12 \pi b^{4}} - \frac {5 x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{4 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erfc}^{2}{\left (b x \right )}}{16 b^{6}} + \frac {11 e^{- 2 b^{2} x^{2}}}{12 \pi b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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