Optimal. Leaf size=126 \[ \frac{(e x)^{m+1} \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac{x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (-\frac{(m+1) \left (4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text{Erfi}\left (\frac{2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1} \]
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Rubi [A] time = 0.322351, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6403, 2278, 2274, 15, 20, 2276, 2234, 2204} \[ \frac{(e x)^{m+1} \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac{x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (-\frac{(m+1) \left (4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text{Erfi}\left (\frac{2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 6403
Rule 2278
Rule 2274
Rule 15
Rule 20
Rule 2276
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \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{(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{(2 b d n) \int \exp \left (a^2 d^2+2 a b d^2 \log \left (c x^n\right )+b^2 d^2 \log ^2\left (c x^n\right )\right ) (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{(2 b d n) \int e^{a^2 d^2+b^2 d^2 \log ^2\left (c x^n\right )} (e x)^m \left (c x^n\right )^{2 a b d^2} \, 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^{-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 a b d^2 n} (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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.441594, size = 126, normalized size = 1. \[ \frac{(e x)^m \left (x \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^{-m} \text{Erfi}\left (\frac{2 a b d^2 n+m+1}{2 b d n}+b d \log \left (c x^n\right )\right ) \exp \left (-\frac{(m+1) \left (4 a b d^2 n+4 b^2 d^2 n \log \left (c x^n\right )-4 b^2 d^2 n^2 \log (x)+m+1\right )}{4 b^2 d^2 n^2}\right )\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\it erfi} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \operatorname{erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.10303, size = 429, normalized size = 3.4 \begin{align*} \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{erfi}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \operatorname{erfi}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \operatorname{erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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