Optimal. Leaf size=39 \[ 2 n^2 p^2 x-2 n p x \log \left (c \left (b x^n\right )^p\right )+x \log ^2\left (c \left (b x^n\right )^p\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2333, 2332,
2495} \begin {gather*} x \log ^2\left (c \left (b x^n\right )^p\right )-2 n p x \log \left (c \left (b x^n\right )^p\right )+2 n^2 p^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2495
Rubi steps
\begin {align*} \int \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\text {Subst}\left (\int \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 )\\ &=x \log ^2\left (c \left (b x^n\right )^p\right )-\text {Subst}\left ((2 n p) \int \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=2 n^2 p^2 x-2 n p x \log \left (c \left (b x^n\right )^p\right )+x \log ^2\left (c \left (b x^n\right )^p\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 37, normalized size = 0.95 \begin {gather*} x \log ^2\left (c \left (b x^n\right )^p\right )-2 n p \left (-n p x+x \log \left (c \left (b x^n\right )^p\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 39, normalized size = 1.00 \begin {gather*} 2 \, n^{2} p^{2} x - 2 \, n p x \log \left (\left (b x^{n}\right )^{p} c\right ) + x \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. 90 vs.
\(2 (39) = 78\).
time = 0.35, size = 90, normalized size = 2.31 \begin {gather*} n^{2} p^{2} x \log \left (x\right )^{2} + 2 \, n^{2} p^{2} x - 2 \, n p^{2} x \log \left (b\right ) + p^{2} x \log \left (b\right )^{2} + x \log \left (c\right )^{2} - 2 \, {\left (n p x - p x \log \left (b\right )\right )} \log \left (c\right ) - 2 \, {\left (n^{2} p^{2} x - n p^{2} x \log \left (b\right ) - n p x \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 = 0.22, size = 39, normalized size = 1.00 \begin {gather*} 2 n^{2} p^{2} x - 2 n p x \log {\left (c \left (b x^{n}\right )^{p} \right )} + x \log {\left (c \left (b x^{n}\right )^{p} \right )}^{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. 92 vs.
\(2 (39) = 78\).
time = 4.76, size = 92, normalized size = 2.36 \begin {gather*} n^{2} p^{2} x \log \left (x\right )^{2} - 2 \, n^{2} p^{2} x \log \left (x\right ) + 2 \, n p^{2} x \log \left (b\right ) \log \left (x\right ) + 2 \, n^{2} p^{2} x - 2 \, n p^{2} x \log \left (b\right ) + p^{2} x \log \left (b\right )^{2} + 2 \, n p x \log \left (c\right ) \log \left (x\right ) - 2 \, n p x \log \left (c\right ) + 2 \, p x \log \left (b\right ) \log \left (c\right ) + x \log \left (c\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.84, size = 39, normalized size = 1.00 \begin {gather*} 2\,x\,n^2\,p^2-2\,x\,n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )+x\,{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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