Optimal. Leaf size=66 \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt{\pi } b d n} \]
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Rubi [A] time = 0.0430606, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {6350} \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt{\pi } b d n} \]
Antiderivative was successfully verified.
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Rule 6350
Rubi steps
\begin{align*} \int \frac{\text{erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{erfc}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \text{erfc}(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n}\\ &=-\frac{e^{-\left (a d+b d \log \left (c x^n\right )\right )^2}}{b d n \sqrt{\pi }}+\frac{\text{erfc}\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.139772, size = 93, normalized size = 1.41 \[ \frac{-\frac{\left (c x^n\right )^{-2 a b d^2} e^{-d^2 \left (a^2+b^2 \log ^2\left (c x^n\right )\right )}}{\sqrt{\pi } b d}-\frac{a \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b}+\log \left (c x^n\right ) \text{Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 80, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ){\it erfc} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) }{n}}+{\frac{{\it erfc} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) a}{bn}}-{\frac{{{\rm e}^{- \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}}{bdn\sqrt{\pi }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00637, size = 80, normalized size = 1.21 \begin{align*} \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname{erfc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac{e^{\left (-{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} d^{2}\right )}}{\sqrt{\pi }}}{b d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13556, size = 308, normalized size = 4.67 \begin{align*} \frac{\pi b d n \log \left (x\right ) -{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname{erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sqrt{\pi } e^{\left (-b^{2} d^{2} n^{2} \log \left (x\right )^{2} - b^{2} d^{2} \log \left (c\right )^{2} - 2 \, a b d^{2} \log \left (c\right ) - a^{2} d^{2} - 2 \,{\left (b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n\right )} \log \left (x\right )\right )}}{\pi b d n} \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 \frac{\operatorname{erfc}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28677, size = 112, normalized size = 1.7 \begin{align*} \frac{b d n \log \left (x\right ) + b d \log \left (c\right ) + a d -{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname{erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \frac{e^{\left (-{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}^{2}\right )}}{\sqrt{\pi }}}{b d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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