Integrand size = 10, antiderivative size = 178 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }+\frac {5 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3 \text {erf}(b x)}{6 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x)^2 \]
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Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6499, 6520, 6508, 30, 2240, 2243} \[ \int x^5 \text {erf}(b x)^2 \, dx=-\frac {5 \text {erf}(b x)^2}{16 b^6}+\frac {x^5 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } b}+\frac {x^4 e^{-2 b^2 x^2}}{6 \pi b^2}+\frac {11 e^{-2 b^2 x^2}}{12 \pi b^6}+\frac {5 x e^{-b^2 x^2} \text {erf}(b x)}{4 \sqrt {\pi } b^5}+\frac {7 x^2 e^{-2 b^2 x^2}}{12 \pi b^4}+\frac {5 x^3 e^{-b^2 x^2} \text {erf}(b x)}{6 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erf}(b x)^2 \]
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Rule 30
Rule 2240
Rule 2243
Rule 6499
Rule 6508
Rule 6520
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \text {erf}(b x)^2-\frac {(2 b) \int e^{-b^2 x^2} x^6 \text {erf}(b x) \, dx}{3 \sqrt {\pi }} \\ & = \frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(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 {erf}(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 {erf}(b x)}{6 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(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 {erf}(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 {erf}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3 \text {erf}(b x)}{6 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(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 {erf}(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 {erf}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3 \text {erf}(b x)}{6 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erf}(b x)^2-\frac {5 \text {Subst}(\int x \, dx,x,\text {erf}(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 {erf}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{-b^2 x^2} x^3 \text {erf}(b x)}{6 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^5 \text {erf}(b x)}{3 b \sqrt {\pi }}-\frac {5 \text {erf}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erf}(b x)^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.60 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2} \left (44+28 b^2 x^2+8 b^4 x^4+4 b e^{b^2 x^2} \sqrt {\pi } x \left (15+10 b^2 x^2+4 b^4 x^4\right ) \text {erf}(b x)+e^{2 b^2 x^2} \pi \left (-15+8 b^6 x^6\right ) \text {erf}(b x)^2\right )}{48 b^6 \pi } \]
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Time = 0.49 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {8 \operatorname {erf}\left (b x \right )^{2} x^{6} b^{6} \pi ^{\frac {3}{2}}+16 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} \operatorname {erf}\left (b x \right ) b^{5} \pi +8 \,{\mathrm e}^{-2 b^{2} x^{2}} x^{4} b^{4} \sqrt {\pi }+40 \,{\mathrm e}^{-b^{2} x^{2}} \operatorname {erf}\left (b x \right ) x^{3} b^{3} \pi +28 x^{2} {\mathrm e}^{-2 b^{2} x^{2}} b^{2} \sqrt {\pi }+60 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erf}\left (b x \right ) b \pi -15 \operatorname {erf}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+44 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{48 b^{6} \pi ^{\frac {3}{2}}}\) | \(160\) |
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.55 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, {\left (2 \, b^{4} x^{4} + 7 \, b^{2} x^{2} + 11\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \]
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Time = 0.54 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.94 \[ \int x^5 \text {erf}(b x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {erf}^{2}{\left (b x \right )}}{6} + \frac {x^{5} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{3 \sqrt {\pi } b} + \frac {x^{4} e^{- 2 b^{2} x^{2}}}{6 \pi b^{2}} + \frac {5 x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{6 \sqrt {\pi } b^{3}} + \frac {7 x^{2} e^{- 2 b^{2} x^{2}}}{12 \pi b^{4}} + \frac {5 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{6}} + \frac {11 e^{- 2 b^{2} x^{2}}}{12 \pi b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int x^5 \text {erf}(b x)^2 \, dx=\int { x^{5} \operatorname {erf}\left (b x\right )^{2} \,d x } \]
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\[ \int x^5 \text {erf}(b x)^2 \, dx=\int { x^{5} \operatorname {erf}\left (b x\right )^{2} \,d x } \]
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Time = 5.66 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80 \[ \int x^5 \text {erf}(b x)^2 \, dx=\frac {x^6\,{\mathrm {erf}\left (b\,x\right )}^2}{6}+\frac {\frac {11\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}-\frac {5\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16}+\frac {7\,b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}+\frac {b^4\,x^4\,{\mathrm {e}}^{-2\,b^2\,x^2}}{6}+\frac {5\,b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{6}+\frac {b^5\,x^5\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}+\frac {5\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4}}{b^6\,\pi } \]
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