Integrand size = 8, antiderivative size = 109 \[ \int x^6 \text {erf}(b x) \, dx=\frac {6 e^{-b^2 x^2}}{7 b^7 \sqrt {\pi }}+\frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erf}(b x) \]
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Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2243, 2240} \[ \int x^6 \text {erf}(b x) \, dx=\frac {x^6 e^{-b^2 x^2}}{7 \sqrt {\pi } b}+\frac {6 e^{-b^2 x^2}}{7 \sqrt {\pi } b^7}+\frac {6 x^2 e^{-b^2 x^2}}{7 \sqrt {\pi } b^5}+\frac {3 x^4 e^{-b^2 x^2}}{7 \sqrt {\pi } b^3}+\frac {1}{7} x^7 \text {erf}(b x) \]
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Rule 2240
Rule 2243
Rule 6496
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} x^7 \text {erf}(b x)-\frac {(2 b) \int e^{-b^2 x^2} x^7 \, dx}{7 \sqrt {\pi }} \\ & = \frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erf}(b x)-\frac {6 \int e^{-b^2 x^2} x^5 \, dx}{7 b \sqrt {\pi }} \\ & = \frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erf}(b x)-\frac {12 \int e^{-b^2 x^2} x^3 \, dx}{7 b^3 \sqrt {\pi }} \\ & = \frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erf}(b x)-\frac {12 \int e^{-b^2 x^2} x \, dx}{7 b^5 \sqrt {\pi }} \\ & = \frac {6 e^{-b^2 x^2}}{7 b^7 \sqrt {\pi }}+\frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}+\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erf}(b x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.66 \[ \int x^6 \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2} \left (6+6 b^2 x^2+3 b^4 x^4+b^6 x^6+b^7 e^{b^2 x^2} \sqrt {\pi } x^7 \text {erf}(b x)\right )}{7 b^7 \sqrt {\pi }} \]
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Time = 0.52 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58
method | result | size |
meijerg | \(\frac {-\frac {12}{7}+\frac {\left (4 b^{6} x^{6}+12 b^{4} x^{4}+24 b^{2} x^{2}+24\right ) {\mathrm e}^{-b^{2} x^{2}}}{14}+\frac {2 x^{7} b^{7} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{7}}{2 b^{7} \sqrt {\pi }}\) | \(63\) |
parallelrisch | \(\frac {x^{7} b^{7} \operatorname {erf}\left (b x \right ) \sqrt {\pi }+{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}+3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}+6 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}+6 \,{\mathrm e}^{-b^{2} x^{2}}}{7 b^{7} \sqrt {\pi }}\) | \(85\) |
derivativedivides | \(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{7} x^{7}}{7}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-3 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-3 \,{\mathrm e}^{-b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}}\) | \(90\) |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{7} x^{7}}{7}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-3 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-3 \,{\mathrm e}^{-b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}}\) | \(90\) |
parts | \(\frac {x^{7} \operatorname {erf}\left (b x \right )}{7}-\frac {2 b \left (-\frac {x^{6} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {3 x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {3 \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}}}{b^{2}}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b^{4}}\right )}{b^{2}}}{b^{2}}\right )}{7 \sqrt {\pi }}\) | \(95\) |
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int x^6 \text {erf}(b x) \, dx=\frac {\pi b^{7} x^{7} \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \]
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Time = 0.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int x^6 \text {erf}(b x) \, dx=\begin {cases} \frac {x^{7} \operatorname {erf}{\left (b x \right )}}{7} + \frac {x^{6} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b} + \frac {3 x^{4} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} + \frac {6 x^{2} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} + \frac {6 e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erf}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erf}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]
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Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erf}(b x) \, dx=\frac {x^7\,\mathrm {erf}\left (b\,x\right )}{7}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,\left (b^6\,x^6+3\,b^4\,x^4+6\,b^2\,x^2+6\right )}{7\,b^7\,\sqrt {\pi }} \]
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