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 = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.13, 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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fricas [B] time = 0.49, size = 128, normalized size = 1.94 \[ \frac {\pi b d n \log \relax (x) - {\left (\pi b d n \log \relax (x) + \pi b d \log \relax (c) + \pi a d\right )} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sqrt {\pi } e^{\left (-b^{2} d^{2} n^{2} \log \relax (x)^{2} - b^{2} d^{2} \log \relax (c)^{2} - 2 \, a b d^{2} \log \relax (c) - a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} n \log \relax (c) + a b d^{2} n\right )} \log \relax (x)\right )}}{\pi b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 83, normalized size = 1.26 \[ \frac {b d n \log \relax (x) + b d \log \relax (c) + a d - {\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )} \operatorname {erf}\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) - \frac {e^{\left (-{\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )}^{2}\right )}}{\sqrt {\pi }}}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 80, normalized size = 1.21 \[ \frac {\ln \left (c \,x^{n}\right ) \mathrm {erfc}\left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n}+\frac {\mathrm {erfc}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) a}{n b}-\frac {{\mathrm e}^{-\left (a d +b d \ln \left (c \,x^{n}\right )\right )^{2}}}{n b d \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 59, normalized size = 0.89 \[ \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {erfc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {e^{\left (-{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} d^{2}\right )}}{\sqrt {\pi }}}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 100, normalized size = 1.52 \[ \frac {\mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{-b^2\,d^2\,{\ln \left (c\,x^n\right )}^2}\,{\mathrm {e}}^{-a^2\,d^2}}{b\,d\,n\,\sqrt {\pi }\,{\left (c\,x^n\right )}^{2\,a\,b\,d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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