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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {\log (c x)}{x}-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) - Log[c*x]/x

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fricas [A]  time = 0.42, size = 11, normalized size = 0.73 \[ -\frac {\log \left (c x\right ) + 1}{x} \]

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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giac [A]  time = 0.21, size = 15, normalized size = 1.00 \[ -\frac {\log \left (c x\right )}{x} - \frac {1}{x} \]

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

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maple [A]  time = 0.03, size = 16, normalized size = 1.07 \[ -\frac {\ln \left (c x \right )}{x}-\frac {1}{x} \]

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 = 0.53, size = 15, normalized size = 1.00 \[ -\frac {\log \left (c x\right )}{x} - \frac {1}{x} \]

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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mupad [B]  time = 3.48, size = 11, normalized size = 0.73 \[ -\frac {\ln \left (c\,x\right )+1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(c*x) + 1)/x

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sympy [A]  time = 0.10, size = 10, normalized size = 0.67 \[ - \frac {\log {\left (c x \right )}}{x} - \frac {1}{x} \]

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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