Integrand size = 19, antiderivative size = 126 \[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {e^{-\frac {(1+m) \left (1+m+4 a b d^2 n\right )}{4 b^2 d^2 n^2}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {1+m+2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )}{2 b d n}\right )}{1+m} \]
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Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6538, 2314, 2308, 2266, 2235} \[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{m+1} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (-\frac {(m+1) \left (4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text {erfi}\left (\frac {2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1} \]
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Rule 2235
Rule 2266
Rule 2308
Rule 2314
Rule 6538
Rubi steps \begin{align*} \text {integral}& = \frac {(e x)^{1+m} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(2 b d n) \int e^{d^2 \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{(1+m) \sqrt {\pi }} \\ & = \frac {(e x)^{1+m} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d n x^{-m-2 a b d^2 n} (e x)^m \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^{m+2 a b d^2 n} \, dx}{(1+m) \sqrt {\pi }} \\ & = \frac {(e x)^{1+m} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d x (e x)^m \left (c x^n\right )^{2 a b d^2-\frac {1+m+2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (a^2 d^2+\frac {\left (1+m+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 )}{(1+m) \sqrt {\pi }} \\ & = \frac {(e x)^{1+m} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d \exp \left (-\frac {(1+m) \left (1+m+4 a b d^2 n\right )}{4 b^2 d^2 n^2}\right ) x (e x)^m \left (c x^n\right )^{2 a b d^2-\frac {1+m+2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {\left (\frac {1+m+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 )}{(1+m) \sqrt {\pi }} \\ & = \frac {(e x)^{1+m} \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\exp \left (-\frac {(1+m) \left (1+m+4 a b d^2 n\right )}{4 b^2 d^2 n^2}\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {1+m+2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )}{2 b d n}\right )}{1+m} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (1+m+4 a b d^2 n-4 b^2 d^2 n^2 \log (x)+4 b^2 d^2 n \log \left (c x^n\right )\right )}{4 b^2 d^2 n^2}} x^{-m} \text {erfi}\left (\frac {1+m+2 a b d^2 n}{2 b d n}+b d \log \left (c x^n\right )\right )\right )}{1+m} \]
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\[\int \left (e x \right )^{m} \operatorname {erfi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.44 \[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + \sqrt {-b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + m + 1\right )} \sqrt {-b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {4 \, b^{2} d^{2} m n^{2} \log \left (e\right ) - 4 \, {\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n \log \left (c\right ) - m^{2} - 4 \, {\left (a b d^{2} m + a b d^{2}\right )} n - 2 \, m - 1}{4 \, b^{2} d^{2} n^{2}}\right )}}{m + 1} \]
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\[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \operatorname {erfi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int (e x)^m \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {erfi}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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