Optimal. Leaf size=94 \[ \frac{1}{2} x^2 \left (c x^n\right )^{-2/n} e^{\frac{1-2 a b d^2 n}{b^2 d^2 n^2}} \text{Erf}\left (\frac{a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac{1}{n}}{b d}\right )+\frac{1}{2} x^2 \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A] time = 0.165809, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {6402, 2278, 2274, 15, 2276, 2234, 2205} \[ \frac{1}{2} x^2 \left (c x^n\right )^{-2/n} e^{\frac{1-2 a b d^2 n}{b^2 d^2 n^2}} \text{Erf}\left (\frac{a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac{1}{n}}{b d}\right )+\frac{1}{2} x^2 \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{(b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} x \, dx}{\sqrt{\pi }}\\ &=\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{(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 \, dx}{\sqrt{\pi }}\\ &=\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{(b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x \left (c x^n\right )^{-2 a b d^2} \, dx}{\sqrt{\pi }}\\ &=\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{\left (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^{1-2 a b d^2 n} \, dx}{\sqrt{\pi }}\\ &=\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{\left (b d x^2 \left (c x^n\right )^{-2 a b d^2-\frac{2-2 a b d^2 n}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-a^2 d^2+\frac{\left (2-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt{\pi }}\\ &=\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac{\left (b d e^{\frac{1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2 a b d^2-\frac{2-2 a b d^2 n}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-\frac{\left (\frac{2-2 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 )}{\sqrt{\pi }}\\ &=\frac{1}{2} e^{\frac{1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text{erf}\left (\frac{a b d^2-\frac{1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right )+\frac{1}{2} x^2 \text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end{align*}
Mathematica [A] time = 0.271673, size = 80, normalized size = 0.85 \[ \frac{1}{2} \left (x^2 e^{\frac{\frac{\frac{1}{d^2}-2 a b n}{b^2}-2 n \log \left (c x^n\right )}{n^2}} \text{Erf}\left (a d+b d \log \left (c x^n\right )-\frac{1}{b d n}\right )+x^2 \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int x{\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 \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 = 2.18754, size = 298, normalized size = 3.17 \begin{align*} -\frac{1}{2} \, x^{2} \operatorname{erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac{1}{2} \, \sqrt{b^{2} d^{2} n^{2}} \operatorname{erf}\left (\frac{{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n - 1\right )} \sqrt{b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac{2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n - 1}{b^{2} d^{2} n^{2}}\right )} + \frac{1}{2} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{erfc}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36917, size = 119, normalized size = 1.27 \begin{align*} -\frac{1}{2} \, x^{2} \operatorname{erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac{1}{2} \, x^{2} - \frac{\operatorname{erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac{1}{b d n}\right ) e^{\left (-\frac{2 \, a}{b n} + \frac{1}{b^{2} d^{2} n^{2}}\right )}}{2 \, c^{\frac{2}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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