3.6 \(\int \frac{\log (c x)}{x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{\log (c x)}{x}-\frac{1}{x} \]

[Out]

-x^(-1) - Log[c*x]/x

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Rubi [A]  time = 0.0071283, antiderivative size = 15, 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} \[ -\frac{\log (c x)}{x}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]/x^2,x]

[Out]

-x^(-1) - Log[c*x]/x

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

Mathematica [A]  time = 0.0008004, size = 15, normalized size = 1. \[ -\frac{\log (c x)}{x}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]/x^2,x]

[Out]

-x^(-1) - Log[c*x]/x

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Maple [A]  time = 0.034, size = 16, normalized size = 1.1 \begin{align*} -{x}^{-1}-{\frac{\ln \left ( cx \right ) }{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x)/x^2,x)

[Out]

-1/x-ln(c*x)/x

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Maxima [A]  time = 0.990233, size = 20, normalized size = 1.33 \begin{align*} -\frac{\log \left (c x\right )}{x} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)/x^2,x, algorithm="maxima")

[Out]

-log(c*x)/x - 1/x

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Fricas [A]  time = 0.860201, size = 26, normalized size = 1.73 \begin{align*} -\frac{\log \left (c x\right ) + 1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)/x^2,x, algorithm="fricas")

[Out]

-(log(c*x) + 1)/x

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Sympy [A]  time = 0.097856, size = 10, normalized size = 0.67 \begin{align*} - \frac{\log{\left (c x \right )}}{x} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x)/x**2,x)

[Out]

-log(c*x)/x - 1/x

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Giac [A]  time = 1.61667, size = 20, normalized size = 1.33 \begin{align*} -\frac{\log \left (c x\right )}{x} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)/x^2,x, algorithm="giac")

[Out]

-log(c*x)/x - 1/x