Optimal. Leaf size=102 \[ \frac{1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac{9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac{3}{n}}{2 b d}\right )+\frac{1}{3} x^3 \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.215069, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {6402, 2278, 2274, 15, 2276, 2234, 2205} \[ \frac{1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac{9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac{3}{n}}{2 b d}\right )+\frac{1}{3} x^3 \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 6402
Rule 2278
Rule 2274
Rule 15
Rule 2276
Rule 2234
Rule 2205
Rubi steps
\begin{align*} \int x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\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{(2 b d n) \int \exp \left (-a^2 d^2-2 a b d^2 \log \left (c x^n\right )-b^2 d^2 \log ^2\left (c x^n\right )\right ) 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{(2 b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^2 \left (c x^n\right )^{-2 a b d^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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.303027, size = 87, normalized size = 0.85 \[ \frac{1}{3} \left (x^3 \text{Erf}\left (a d+b d \log \left (c x^n\right )-\frac{3}{2 b d n}\right ) \exp \left (\frac{3 \left (\frac{\frac{3}{d^2}-4 a b n}{b^2}-4 n \log \left (c x^n\right )\right )}{4 n^2}\right )+x^3 \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.208, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\it erfc} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18052, size = 317, normalized size = 3.11 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34418, size = 122, normalized size = 1.2 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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