Integrand size = 17, antiderivative size = 102 \[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{3} e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-3/n} \text {erf}\left (\frac {2 a b d^2-\frac {3}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )+\frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6537, 2314, 2308, 2266, 2236} \[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text {erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {3}{n}}{2 b d}\right )+\frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rule 2236
Rule 2266
Rule 2308
Rule 2314
Rule 6537
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {(2 b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx}{3 \sqrt {\pi }} \\ & = \frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (2 b d n x^{2 a b d^2 n} \left (c x^n\right )^{-2 a b d^2}\right ) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^{2-2 a b d^2 n} \, dx}{3 \sqrt {\pi }} \\ & = \frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (2 b d x^3 \left (c x^n\right )^{-2 a b d^2-\frac {3-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (3-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{3 \sqrt {\pi }} \\ & = \frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (2 b d e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-2 a b d^2-\frac {3-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {\left (\frac {3-2 a b d^2 n}{n}-2 b^2 d^2 x\right )^2}{4 b^2 d^2}\right ) \, dx,x,\log \left (c x^n\right )\right )}{3 \sqrt {\pi }} \\ & = \frac {1}{3} e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-3/n} \text {erf}\left (\frac {2 a b d^2-\frac {3}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )+\frac {1}{3} x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.85 \[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{3} \left (e^{\frac {3 \left (\frac {\frac {3}{d^2}-4 a b n}{b^2}-4 n \log \left (c x^n\right )\right )}{4 n^2}} x^3 \text {erf}\left (a d-\frac {3}{2 b d n}+b d \log \left (c x^n\right )\right )+x^3 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]
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\[\int x^{2} \operatorname {erfc}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.27 \[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{3} \, x^{3} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{3} \, x^{3} + \frac {1}{3} \, \sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n - 3\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {3 \, {\left (4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n - 3\right )}}{4 \, b^{2} d^{2} n^{2}}\right )} \]
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\[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \operatorname {erfc}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \operatorname {erfc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{3} \, x^{3} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac {1}{3} \, x^{3} - \frac {\operatorname {erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac {3}{2 \, b d n}\right ) e^{\left (-\frac {3 \, a}{b n} + \frac {9}{4 \, b^{2} d^{2} n^{2}}\right )}}{3 \, c^{\frac {3}{n}}} \]
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Timed out. \[ \int x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,\mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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