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]

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

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Rubi [A]  time = 0.0074127, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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*}

Mathematica [A]  time = 0.0008967, size = 15, normalized size = 1. \[ \frac{\log ^2\left (c x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}}{2\,n}} \end{align*}

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 = 1.00267, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (c x^{n}\right )^{2}}{2 \, n} \end{align*}

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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Fricas [A]  time = 1.83902, size = 43, normalized size = 2.87 \begin{align*} \frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (c\right ) \log \left (x\right ) \end{align*}

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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Sympy [A]  time = 1.68928, size = 51, normalized size = 3.4 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.30343, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (c\right ) \log \left (x\right ) \end{align*}

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)