Optimal. Leaf size=94 \[ \frac {1}{2} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text {Erf}\left (\frac {a b d^2-\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right )+\frac {1}{2} x^2 \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6537, 2314,
2308, 2266, 2236} \begin {gather*} \frac {1}{2} x^2 \left (c x^n\right )^{-2/n} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} \text {Erf}\left (\frac {a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{b d}\right )+\frac {1}{2} x^2 \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 2266
Rule 2308
Rule 2314
Rule 6537
Rubi steps
\begin {align*} \int x \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {(b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} x \, dx}{\sqrt {\pi }}\\ &=\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {(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 \, dx}{\sqrt {\pi }}\\ &=\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {(b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x \left (c x^n\right )^{-2 a b d^2} \, dx}{\sqrt {\pi }}\\ &=\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (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^{1-2 a b d^2 n} \, dx}{\sqrt {\pi }}\\ &=\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (b d x^2 \left (c x^n\right )^{-2 a b d^2-\frac {2-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (2-2 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 }}\\ &=\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (b d e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2 a b d^2-\frac {2-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {\left (\frac {2-2 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 }}\\ &=\frac {1}{2} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text {erf}\left (\frac {a b d^2-\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right )+\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 80, normalized size = 0.85 \begin {gather*} \frac {1}{2} \left (e^{\frac {\frac {\frac {1}{d^2}-2 a b n}{b^2}-2 n \log \left (c x^n\right )}{n^2}} x^2 \text {Erf}\left (a d-\frac {1}{b d n}+b d \log \left (c x^n\right )\right )+x^2 \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int x \,\mathrm {erfc}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 126, normalized size = 1.34 \begin {gather*} -\frac {1}{2} \, x^{2} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{2} \, \sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n - 1}{b^{2} d^{2} n^{2}}\right )} + \frac {1}{2} \, x^{2} \end {gather*}
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 \begin {gather*} \int x \operatorname {erfc}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 88, normalized size = 0.94 \begin {gather*} -\frac {1}{2} \, x^{2} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac {1}{2} \, x^{2} - \frac {\operatorname {erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac {1}{b d n}\right ) e^{\left (-\frac {2 \, a}{b n} + \frac {1}{b^{2} d^{2} n^{2}}\right )}}{2 \, c^{\frac {2}{n}}} \end {gather*}
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 \begin {gather*} \int x\,\mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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