Optimal. Leaf size=102 \[ \frac {1}{3} x^3 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text {erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {3}{n}}{2 b d}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6401, 2278, 2274, 15, 2276, 2234, 2205} \[ \frac {1}{3} x^3 \text {Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text {Erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {3}{n}}{2 b d}\right ) \]
Antiderivative was successfully verified.
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Rule 15
Rule 2205
Rule 2234
Rule 2274
Rule 2276
Rule 2278
Rule 6401
Rubi steps
\begin {align*} \int x^2 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {1}{3} x^3 \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} x^2 \, dx}{3 \sqrt {\pi }}\\ &=\frac {1}{3} x^3 \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 ) x^2 \, dx}{3 \sqrt {\pi }}\\ &=\frac {1}{3} x^3 \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 )} x^2 \left (c x^n\right )^{-2 a b d^2} \, dx}{3 \sqrt {\pi }}\\ &=\frac {1}{3} x^3 \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^{2-2 a b d^2 n} \, dx}{3 \sqrt {\pi }}\\ &=\frac {1}{3} x^3 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d x^3 \left (c x^n\right )^{-2 a b d^2-\frac {3-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (3-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{3 \sqrt {\pi }}\\ &=\frac {1}{3} x^3 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-2 a b d^2-\frac {3-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-\frac {\left (\frac {3-2 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 )}{3 \sqrt {\pi }}\\ &=\frac {1}{3} x^3 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} e^{\frac {9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-3/n} \text {erf}\left (\frac {2 a b d^2-\frac {3}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )\\ \end {align*}
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Mathematica [A] time = 0.35, size = 88, normalized size = 0.86 \[ \frac {1}{3} \left (x^3 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^3 \text {erf}\left (a d+b d \log \left (c x^n\right )-\frac {3}{2 b d n}\right ) \exp \left (-\frac {3 \left (\frac {4 a b n-\frac {3}{d^2}}{b^2}+4 n \log \left (c x^n\right )\right )}{4 n^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 125, normalized size = 1.23 \[ \frac {1}{3} \, x^{3} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{3} \, \sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \relax (x) + 2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n - 3\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {3 \, {\left (4 \, b^{2} d^{2} n \log \relax (c) + 4 \, a b d^{2} n - 3\right )}}{4 \, b^{2} d^{2} n^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 85, normalized size = 0.83 \[ \frac {1}{3} \, x^{3} \operatorname {erf}\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) + \frac {\operatorname {erf}\left (-b d n \log \relax (x) - b d \log \relax (c) - a d + \frac {3}{2 \, b d n}\right ) e^{\left (-\frac {3 \, a}{b n} + \frac {9}{4 \, b^{2} d^{2} n^{2}}\right )}}{3 \, c^{\frac {3}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int x^{2} \erf \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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