Optimal. Leaf size=27 \[ -\frac{a+b \log \left (c x^n\right )}{2 x^2}-\frac{b n}{4 x^2} \]
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Rubi [A] time = 0.0125244, antiderivative size = 27, 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} \[ -\frac{a+b \log \left (c x^n\right )}{2 x^2}-\frac{b n}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx &=-\frac{b n}{4 x^2}-\frac{a+b \log \left (c x^n\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0012246, size = 32, normalized size = 1.19 \[ -\frac{a}{2 x^2}-\frac{b \log \left (c x^n\right )}{2 x^2}-\frac{b n}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.073, size = 111, normalized size = 4.1 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{2\,{x}^{2}}}-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +bn+2\,a}{4\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13792, size = 35, normalized size = 1.3 \begin{align*} -\frac{b n}{4 \, x^{2}} - \frac{b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.834715, size = 68, normalized size = 2.52 \begin{align*} -\frac{2 \, b n \log \left (x\right ) + b n + 2 \, b \log \left (c\right ) + 2 \, a}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2403, size = 37, normalized size = 1.37 \begin{align*} - \frac{a}{2 x^{2}} - \frac{b n \log{\left (x \right )}}{2 x^{2}} - \frac{b n}{4 x^{2}} - \frac{b \log{\left (c \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14973, size = 36, normalized size = 1.33 \begin{align*} -\frac{b n \log \left (x\right )}{2 \, x^{2}} - \frac{b n + 2 \, b \log \left (c\right ) + 2 \, a}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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