3.7 \(\int \frac{\log (c x)}{x^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]

[Out]

-1/(4*x^2) - Log[c*x]/(2*x^2)

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Rubi [A]  time = 0.0066814, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2304} \[ -\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]/x^3,x]

[Out]

-1/(4*x^2) - Log[c*x]/(2*x^2)

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{\log (c x)}{x^3} \, dx &=-\frac{1}{4 x^2}-\frac{\log (c x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0008228, size = 19, normalized size = 1. \[ -\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]/x^3,x]

[Out]

-1/(4*x^2) - Log[c*x]/(2*x^2)

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Maple [A]  time = 0.035, size = 16, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{2}}}-{\frac{\ln \left ( cx \right ) }{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x)/x^3,x)

[Out]

-1/4/x^2-1/2*ln(c*x)/x^2

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Maxima [A]  time = 0.97228, size = 20, normalized size = 1.05 \begin{align*} -\frac{\log \left (c x\right )}{2 \, x^{2}} - \frac{1}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)/x^3,x, algorithm="maxima")

[Out]

-1/2*log(c*x)/x^2 - 1/4/x^2

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Fricas [A]  time = 0.826076, size = 36, normalized size = 1.89 \begin{align*} -\frac{2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)/x^3,x, algorithm="fricas")

[Out]

-1/4*(2*log(c*x) + 1)/x^2

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Sympy [A]  time = 0.131911, size = 17, normalized size = 0.89 \begin{align*} - \frac{\log{\left (c x \right )}}{2 x^{2}} - \frac{1}{4 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x)/x**3,x)

[Out]

-log(c*x)/(2*x**2) - 1/(4*x**2)

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Giac [A]  time = 1.09793, size = 20, normalized size = 1.05 \begin{align*} -\frac{\log \left (c x\right )}{2 \, x^{2}} - \frac{1}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)/x^3,x, algorithm="giac")

[Out]

-1/2*log(c*x)/x^2 - 1/4/x^2