Optimal. Leaf size=39 \[ x \log ^2\left (c \left (b x^n\right )^p\right )-2 n p x \log \left (c \left (b x^n\right )^p\right )+2 n^2 p^2 x \]
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Rubi [A] time = 0.0220322, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2296, 2295, 2445} \[ x \log ^2\left (c \left (b x^n\right )^p\right )-2 n p x \log \left (c \left (b x^n\right )^p\right )+2 n^2 p^2 x \]
Antiderivative was successfully verified.
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Rule 2296
Rule 2295
Rule 2445
Rubi steps
\begin{align*} \int \log ^2\left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{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 )-\operatorname{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*}
Mathematica [A] time = 0.0026566, size = 37, normalized size = 0.95 \[ x \log ^2\left (c \left (b x^n\right )^p\right )-2 n p \left (x \log \left (c \left (b x^n\right )^p\right )-n p x\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2046, size = 53, normalized size = 1.36 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.926268, size = 230, normalized size = 5.9 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.17567, size = 116, normalized size = 2.97 \begin{align*} n^{2} p^{2} x \log{\left (x \right )}^{2} - 2 n^{2} p^{2} x \log{\left (x \right )} + 2 n^{2} p^{2} x + 2 n p^{2} x \log{\left (b \right )} \log{\left (x \right )} - 2 n p^{2} x \log{\left (b \right )} + 2 n p x \log{\left (c \right )} \log{\left (x \right )} - 2 n p x \log{\left (c \right )} + p^{2} x \log{\left (b \right )}^{2} + 2 p x \log{\left (b \right )} \log{\left (c \right )} + x \log{\left (c \right )}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3223, size = 124, normalized size = 3.18 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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