Optimal. Leaf size=52 \[ \frac {1}{4} n^2 p^2 x^2-\frac {1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac {1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2342, 2341,
2495} \begin {gather*} \frac {1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\frac {1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac {1}{4} n^2 p^2 x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2342
Rule 2495
Rubi steps
\begin {align*} \int x \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\text {Subst}\left (\int x \log ^2\left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac {1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )-\text {Subst}\left ((n p) \int x \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac {1}{4} n^2 p^2 x^2-\frac {1}{2} n p x^2 \log \left (c \left (b x^n\right )^p\right )+\frac {1}{2} x^2 \log ^2\left (c \left (b x^n\right )^p\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.83 \begin {gather*} \frac {1}{4} x^2 \left (n^2 p^2-2 n p \log \left (c \left (b x^n\right )^p\right )+2 \log ^2\left (c \left (b x^n\right )^p\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 46, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, n^{2} p^{2} x^{2} - \frac {1}{2} \, n p x^{2} \log \left (\left (b x^{n}\right )^{p} c\right ) + \frac {1}{2} \, x^{2} \log \left (\left (b x^{n}\right )^{p} c\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs.
\(2 (46) = 92\).
time = 0.34, size = 113, normalized size = 2.17 \begin {gather*} \frac {1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right )^{2} + \frac {1}{4} \, n^{2} p^{2} x^{2} - \frac {1}{2} \, n p^{2} x^{2} \log \left (b\right ) + \frac {1}{2} \, p^{2} x^{2} \log \left (b\right )^{2} + \frac {1}{2} \, x^{2} \log \left (c\right )^{2} - \frac {1}{2} \, {\left (n p x^{2} - 2 \, p x^{2} \log \left (b\right )\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n^{2} p^{2} x^{2} - 2 \, n p^{2} x^{2} \log \left (b\right ) - 2 \, n p x^{2} \log \left (c\right )\right )} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 46, normalized size = 0.88 \begin {gather*} \frac {n^{2} p^{2} x^{2}}{4} - \frac {n p x^{2} \log {\left (c \left (b x^{n}\right )^{p} \right )}}{2} + \frac {x^{2} \log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (46) = 92\).
time = 7.38, size = 112, normalized size = 2.15 \begin {gather*} \frac {1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right )^{2} - \frac {1}{2} \, n^{2} p^{2} x^{2} \log \left (x\right ) + n p^{2} x^{2} \log \left (b\right ) \log \left (x\right ) + \frac {1}{4} \, n^{2} p^{2} x^{2} - \frac {1}{2} \, n p^{2} x^{2} \log \left (b\right ) + \frac {1}{2} \, p^{2} x^{2} \log \left (b\right )^{2} + n p x^{2} \log \left (c\right ) \log \left (x\right ) - \frac {1}{2} \, n p x^{2} \log \left (c\right ) + p x^{2} \log \left (b\right ) \log \left (c\right ) + \frac {1}{2} \, x^{2} \log \left (c\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.85, size = 46, normalized size = 0.88 \begin {gather*} \frac {n^2\,p^2\,x^2}{4}-\frac {n\,p\,x^2\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{2}+\frac {x^2\,{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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