Optimal. Leaf size=18 \[ -n p x+x \log \left (c \left (b x^n\right )^p\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2332, 2495}
\begin {gather*} x \log \left (c \left (b x^n\right )^p\right )-n p x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2495
Rubi steps
\begin {align*} \int \log \left (c \left (b x^n\right )^p\right ) \, dx &=\text {Subst}\left (\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 )\\ &=-n p x+x \log \left (c \left (b x^n\right )^p\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -n p x+x \log \left (c \left (b x^n\right )^p\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 1.06
method | result | size |
default | \(-n p x +x \ln \left (c \left (b \,x^{n}\right )^{p}\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 18, normalized size = 1.00 \begin {gather*} -n p x + x \log \left (\left (b x^{n}\right )^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 21, normalized size = 1.17 \begin {gather*} n p x \log \left (x\right ) - n p x + p x \log \left (b\right ) + x \log \left (c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 15, normalized size = 0.83 \begin {gather*} - n p x + x \log {\left (c \left (b x^{n}\right )^{p} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.82, size = 21, normalized size = 1.17 \begin {gather*} n p x \log \left (x\right ) - n p x + p x \log \left (b\right ) + x \log \left (c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 0.94 \begin {gather*} x\,\left (\ln \left (c\,{\left (b\,x^n\right )}^p\right )-n\,p\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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