Optimal. Leaf size=91 \[ x \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{-\frac{4 a b d^2 n+1}{4 b^2 d^2 n^2}} \text{Erfi}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}}{2 b d}\right ) \]
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Rubi [A] time = 0.132879, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {6399, 2277, 2274, 15, 2276, 2234, 2204} \[ x \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{-\frac{4 a b d^2 n+1}{4 b^2 d^2 n^2}} \text{Erfi}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}}{2 b d}\right ) \]
Antiderivative was successfully verified.
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Rule 6399
Rule 2277
Rule 2274
Rule 15
Rule 2276
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{(2 b d n) \int e^{d^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{\sqrt{\pi }}\\ &=x \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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 ) \, dx}{\sqrt{\pi }}\\ &=x \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{(2 b d n) \int e^{a^2 d^2+b^2 d^2 \log ^2\left (c x^n\right )} \left (c x^n\right )^{2 a b d^2} \, dx}{\sqrt{\pi }}\\ &=x \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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} \, dx}{\sqrt{\pi }}\\ &=x \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{\left (2 b d x \left (c x^n\right )^{2 a b d^2-\frac{1+2 a b d^2 n}{n}}\right ) \operatorname{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 \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{\left (2 b d e^{-\frac{1+4 a b d^2 n}{4 b^2 d^2 n^2}} x \left (c x^n\right )^{2 a b d^2-\frac{1+2 a b d^2 n}{n}}\right ) \operatorname{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 \text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac{1+4 a b d^2 n}{4 b^2 d^2 n^2}} x \left (c x^n\right )^{-1/n} \text{erfi}\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 )\\ \end{align*}
Mathematica [A] time = 0.236883, size = 78, normalized size = 0.86 \[ x \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \text{Erfi}\left (a d+b d \log \left (c x^n\right )+\frac{1}{2 b d n}\right ) \exp \left (-\frac{\frac{4 a b n+\frac{1}{d^2}}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\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 \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 = 3.38574, size = 293, normalized size = 3.22 \begin{align*} -\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 + 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 )} + x \operatorname{erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) \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 \operatorname{erfi}{\left (d \left (a + b \log{\left (c x^{n} \right )}\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 \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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