Integrand size = 15, antiderivative size = 93 \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {a b d^2+\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6538, 2314, 2308, 2266, 2235} \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x^2 \left (c x^n\right )^{-2/n} e^{-\frac {2 a b d^2 n+1}{b^2 d^2 n^2}} \text {erfi}\left (\frac {a b d^2+b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}}{b d}\right ) \]
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Rule 2235
Rule 2266
Rule 2308
Rule 2314
Rule 6538
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {(b d n) \int e^{d^2 \left (a+b \log \left (c x^n\right )\right )^2} x \, dx}{\sqrt {\pi }} \\ & = \frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (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^{1+2 a b d^2 n} \, dx}{\sqrt {\pi }} \\ & = \frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (b d x^2 \left (c x^n\right )^{2 a b d^2-\frac {2+2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (a^2 d^2+\frac {\left (2+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 }} \\ & = \frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (b d e^{-\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{2 a b d^2-\frac {2+2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {\left (\frac {2+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 }} \\ & = \frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {a b d^2+\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \left (x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {\frac {\frac {1}{d^2}+2 a b n}{b^2}+2 n \log \left (c x^n\right )}{n^2}} x^2 \text {erfi}\left (a d+\frac {1}{b d n}+b d \log \left (c x^n\right )\right )\right ) \]
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\[\int x \,\operatorname {erfi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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none
Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.32 \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{2} \, \sqrt {-b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n + 1\right )} \sqrt {-b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + 1}{b^{2} d^{2} n^{2}}\right )} \]
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\[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x \operatorname {erfi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x\,\mathrm {erfi}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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