3.12 \(\int \frac {\log (c x^n)}{x} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log ^2\left (c x^n\right )}{2 n} \]

[Out]

1/2*ln(c*x^n)^2/n

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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2301} \[ \frac {\log ^2\left (c x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c*x^n]^2/(2*n)

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {align*} \int \frac {\log \left (c x^n\right )}{x} \, dx &=\frac {\log ^2\left (c x^n\right )}{2 n}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log ^2\left (c x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c*x^n]^2/(2*n)

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fricas [A]  time = 0.43, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, n \log \relax (x)^{2} + \log \relax (c) \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*n*log(x)^2 + log(c)*log(x)

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giac [A]  time = 0.16, size = 13, normalized size = 0.87 \[ \frac {1}{2} \, n \log \relax (x)^{2} + \log \relax (c) \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*n*log(x)^2 + log(c)*log(x)

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maple [A]  time = 0.07, size = 14, normalized size = 0.93 \[ \frac {\ln \left (c \,x^{n}\right )^{2}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(c*x^n)^2/n

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maxima [A]  time = 0.73, size = 13, normalized size = 0.87 \[ \frac {\log \left (c x^{n}\right )^{2}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(c*x^n)^2/n

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mupad [B]  time = 0.25, size = 13, normalized size = 0.87 \[ \frac {{\ln \left (c\,x^n\right )}^2}{2\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^n)^2/(2*n)

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sympy [A]  time = 1.70, size = 51, normalized size = 3.40 \[ \begin {cases} \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{2}}{2 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {{G_{3, 3}^{3, 0}\left (\begin {matrix} & 1, 1, 1 \\0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {{G_{3, 3}^{0, 3}\left (\begin {matrix} 1, 1, 1 & \\ & 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(c*x**n)**2/(2*n), Abs(c*x**n) < 1), (log(x**(-n)/c)**2/(2*n), 1/Abs(c*x**n) < 1), (meijerg(((),
(1, 1, 1)), ((0, 0, 0), ()), c*x**n)/n + meijerg(((1, 1, 1), ()), ((), (0, 0, 0)), c*x**n)/n, True))

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