Integrand size = 17, antiderivative size = 66 \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{b d n \sqrt {\pi }}+\frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6485} \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt {\pi } b d n} \]
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Rule 6485
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {erfc}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {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*}
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.41 \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {-\frac {e^{-d^2 \left (a^2+b^2 \log ^2\left (c x^n\right )\right )} \left (c x^n\right )^{-2 a b d^2}}{b d \sqrt {\pi }}-\frac {a \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b}+\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n} \]
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Time = 1.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \operatorname {erfc}\left (a d +b d \ln \left (c \,x^{n}\right )\right )-\frac {{\mathrm e}^{-{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\) | \(63\) |
default | \(\frac {\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \operatorname {erfc}\left (a d +b d \ln \left (c \,x^{n}\right )\right )-\frac {{\mathrm e}^{-{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\) | \(63\) |
parts | \(\ln \left (x \right ) \operatorname {erfc}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+\frac {2 d b n \left (-\frac {{\mathrm e}^{-\ln \left (x \right )^{2} b^{2} d^{2} n^{2}-2 d^{2} \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) n b \ln \left (x \right )-d^{2} {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )}^{2}}}{2 b^{2} d^{2} n^{2}}-\frac {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) \sqrt {\pi }\, \operatorname {erf}\left (d b n \ln \left (x \right )+d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{2 d \,n^{2} b^{2}}\right )}{\sqrt {\pi }}\) | \(159\) |
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (63) = 126\).
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.94 \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\pi b d n \log \left (x\right ) - {\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sqrt {\pi } e^{\left (-b^{2} d^{2} n^{2} \log \left (x\right )^{2} - b^{2} d^{2} \log \left (c\right )^{2} - 2 \, a b d^{2} \log \left (c\right ) - a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n\right )} \log \left (x\right )\right )}}{\pi b d n} \]
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\[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {erfc}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {erfc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {e^{\left (-{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} d^{2}\right )}}{\sqrt {\pi }}}{b d n} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26 \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {b d n \log \left (x\right ) + b d \log \left (c\right ) + a d - {\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \frac {e^{\left (-{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}^{2}\right )}}{\sqrt {\pi }}}{b d n} \]
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Time = 5.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.52 \[ \int \frac {\text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{-b^2\,d^2\,{\ln \left (c\,x^n\right )}^2}\,{\mathrm {e}}^{-a^2\,d^2}}{b\,d\,n\,\sqrt {\pi }\,{\left (c\,x^n\right )}^{2\,a\,b\,d^2}} \]
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