Integrand size = 17, antiderivative size = 92 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {2 a b d^2+\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x} \]
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Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6536, 2314, 2308, 2266, 2236} \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\left (c x^n\right )^{\frac {1}{n}} e^{\frac {a}{b n}+\frac {1}{4 b^2 d^2 n^2}} \text {erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}}{2 b d}\right )}{x}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
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Rule 2236
Rule 2266
Rule 2308
Rule 2314
Rule 6536
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {(2 b d n) \int \frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 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^{-2-2 a b d^2 n} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d \left (c x^n\right )^{-2 a b d^2-\frac {-1-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (-1-2 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 } x} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d e^{\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{-2 a b d^2-\frac {-1-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {\left (\frac {-1-2 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 } x} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {2 a b d^2+\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {-\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {\frac {\frac {1}{d^2}+4 a b n}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}} \text {erf}\left (a d+\frac {1}{2 b d n}+b d \log \left (c x^n\right )\right )}{x} \]
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\[\int \frac {\operatorname {erf}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.37 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\sqrt {b^{2} d^{2} n^{2}} x \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n + 1}{4 \, b^{2} d^{2} n^{2}}\right )} - \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x} \]
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\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]
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