Integrand size = 17, antiderivative size = 65 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{b d n \sqrt {\pi }}+\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Time = 0.04 (sec) , antiderivative size = 65, 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 = {6484} \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt {\pi } b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rule 6484
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {erf}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {Subst}\left (\int \text {erf}(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 {erf}\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 = 79, normalized size of antiderivative = 1.22 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\frac {e^{-d^2 \left (a^2+b^2 \log ^2\left (c x^n\right )\right )} \left (c x^n\right )^{-2 a b d^2}}{b d \sqrt {\pi }}+\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (\frac {a}{b}+\log \left (c x^n\right )\right )}{n} \]
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Time = 1.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\operatorname {erf}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {{\mathrm e}^{-{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\) | \(62\) |
default | \(\frac {\operatorname {erf}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {{\mathrm e}^{-{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\) | \(62\) |
parts | \(\ln \left (x \right ) \operatorname {erf}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {2 d b n \left (-\frac {{\mathrm e}^{-\ln \left (x \right )^{2} b^{2} d^{2} n^{2}-2 d^{2} \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) n b \ln \left (x \right )-d^{2} {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )}^{2}}}{2 b^{2} d^{2} n^{2}}-\frac {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) \sqrt {\pi }\, \operatorname {erf}\left (d b n \ln \left (x \right )+d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{2 d \,n^{2} b^{2}}\right )}{\sqrt {\pi }}\) | \(159\) |
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Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.83 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \sqrt {\pi } e^{\left (-b^{2} d^{2} n^{2} \log \left (x\right )^{2} - b^{2} d^{2} \log \left (c\right )^{2} - 2 \, a b d^{2} \log \left (c\right ) - a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n\right )} \log \left (x\right )\right )}}{\pi b d n} \]
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\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) + \frac {e^{\left (-{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} d^{2}\right )}}{\sqrt {\pi }}}{b d n} \]
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Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac {e^{\left (-{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}^{2}\right )}}{\sqrt {\pi }}}{b d n} \]
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Time = 5.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.86 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\ln \left (c\,x^n\right )\,\mathrm {erf}\left (a\,d+b\,d\,\ln \left (c\,x^n\right )\right )}{n}+\frac {a\,d\,\mathrm {erfi}\left (a\,\sqrt {-d^2}+b\,\ln \left (c\,x^n\right )\,\sqrt {-d^2}\right )}{b\,n\,\sqrt {-d^2}}+\frac {{\mathrm {e}}^{-b^2\,d^2\,{\ln \left (c\,x^n\right )}^2}\,{\mathrm {e}}^{-a^2\,d^2}}{b\,d\,n\,\sqrt {\pi }\,{\left (c\,x^n\right )}^{2\,a\,b\,d^2}} \]
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