Optimal. Leaf size=93 \[ x \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{\frac{1-4 a b d^2 n}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac{1}{n}}{2 b d}\right ) \]
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Rubi [A] time = 0.128592, antiderivative size = 93, 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 = {6397, 2277, 2274, 15, 2276, 2234, 2205} \[ x \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{\frac{1-4 a b d^2 n}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac{1}{n}}{2 b d}\right ) \]
Antiderivative was successfully verified.
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Rule 6397
Rule 2277
Rule 2274
Rule 15
Rule 2276
Rule 2234
Rule 2205
Rubi steps
\begin{align*} \int \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{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 )\\ \end{align*}
Mathematica [A] time = 0.244979, size = 80, normalized size = 0.86 \[ x \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \text{Erf}\left (a d+b d \log \left (c x^n\right )-\frac{1}{2 b d n}\right ) \exp \left (-\frac{\frac{4 a b n-\frac{1}{d^2}}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{\it Erf} \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*} -\frac{2 \, b d n e^{\left (-b^{2} d^{2} \log \left (c\right )^{2} - a^{2} d^{2}\right )} \int \frac{e^{\left (-2 \, b^{2} d^{2} \log \left (c\right ) \log \left (x^{n}\right ) - b^{2} d^{2} \log \left (x^{n}\right )^{2}\right )}}{{\left (x^{n}\right )}^{2 \, a b d^{2}}}\,{d x}}{\sqrt{\pi } c^{2 \, a b d^{2}}} + x \operatorname{erf}\left (b d \log \left (x^{n}\right ) +{\left (b \log \left (c\right ) + a\right )} d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.82209, size = 290, normalized size = 3.12 \begin{align*} -\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 - 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{erf}\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{erf}{\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 [A] time = 1.34115, size = 107, normalized size = 1.15 \begin{align*} x \operatorname{erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac{\operatorname{erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac{1}{2 \, b d n}\right ) e^{\left (-\frac{a}{b n} + \frac{1}{4 \, b^{2} d^{2} n^{2}}\right )}}{c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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