Optimal. Leaf size=32 \[ \frac{(d x)^n \log \left (c x^n\right )}{d n}-\frac{(d x)^n}{d n} \]
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Rubi [A] time = 0.0110705, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2304} \[ \frac{(d x)^n \log \left (c x^n\right )}{d n}-\frac{(d x)^n}{d n} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin{align*} \int (d x)^{-1+n} \log \left (c x^n\right ) \, dx &=-\frac{(d x)^n}{d n}+\frac{(d x)^n \log \left (c x^n\right )}{d n}\\ \end{align*}
Mathematica [A] time = 0.0041659, size = 20, normalized size = 0.62 \[ \frac{(d x)^n \left (\log \left (c x^n\right )-1\right )}{d n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.088, size = 263, normalized size = 8.2 \begin{align*}{\frac{x\ln \left ({x}^{n} \right ) }{n}{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}}+{\frac{ \left ( i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( c \right ) -2 \right ) x}{2\,n}{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.993306, size = 55, normalized size = 1.72 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (c\right ) - 1\right )} d^{n - 1} x^{n}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 39.6049, size = 68, normalized size = 2.12 \begin{align*} \begin{cases} \tilde{\infty } x \log{\left (c \right )} & \text{for}\: d = 0 \wedge n = 0 \\\frac{\log{\left (c \right )} \log{\left (x \right )}}{d} & \text{for}\: n = 0 \\0^{n - 1} \left (n x \log{\left (x \right )} - n x + x \log{\left (c \right )}\right ) & \text{for}\: d = 0 \\\frac{d^{n} x^{n} \log{\left (x \right )}}{d} + \frac{d^{n} x^{n} \log{\left (c \right )}}{d n} - \frac{d^{n} x^{n}}{d n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38578, size = 57, normalized size = 1.78 \begin{align*} \frac{d^{n} x^{n} \log \left (x\right )}{d} + \frac{d^{n} x^{n} \log \left (c\right )}{d n} - \frac{d^{n} x^{n}}{d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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