Optimal. Leaf size=22 \[ \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2301, 2445} \[ \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2445
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (b x^n\right )^p\right )}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\log \left (b^p c x^{n p}\right )}{x} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 n p} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 19, normalized size = 0.86 \[ \frac {1}{2} \, n p \log \relax (x)^{2} + {\left (p \log \relax (b) + \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 20, normalized size = 0.91 \[ \frac {1}{2} \, n p \log \relax (x)^{2} + p \log \relax (b) \log \relax (x) + \log \relax (c) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.95 \[ \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{2 n p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 20, normalized size = 0.91 \[ \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{2 \, n p} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.80, size = 20, normalized size = 0.91 \[ \frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{2\,n\,p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.02, size = 37, normalized size = 1.68 \[ - \begin {cases} - \log {\relax (x )} \log {\left (b^{p} c \right )} & \text {for}\: n = 0 \\- \log {\relax (c )} \log {\relax (x )} & \text {for}\: p = 0 \\- \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{2 n p} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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