3.47 \(\int \frac {a+b \log (c x^n)}{x} \, dx\)

Optimal. Leaf size=22 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

[Out]

1/2*(a+b*ln(c*x^n))^2/b/n

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2301} \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/x,x]

[Out]

(a + b*Log[c*x^n])^2/(2*b*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 {a+b \log \left (c x^n\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.95 \[ a \log (x)+\frac {b \log ^2\left (c x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/x,x]

[Out]

a*Log[x] + (b*Log[c*x^n]^2)/(2*n)

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fricas [A]  time = 0.45, size = 18, normalized size = 0.82 \[ \frac {1}{2} \, b n \log \relax (x)^{2} + {\left (b \log \relax (c) + a\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*n*log(x)^2 + (b*log(c) + a)*log(x)

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giac [A]  time = 0.22, size = 19, normalized size = 0.86 \[ \frac {1}{2} \, b n \log \relax (x)^{2} + b \log \relax (c) \log \relax (x) + a \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*n*log(x)^2 + b*log(c)*log(x) + a*log(x)

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maple [A]  time = 0.03, size = 27, normalized size = 1.23 \[ \frac {b \ln \left (c \,x^{n}\right )^{2}}{2 n}+\frac {a \ln \left (c \,x^{n}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x,x)

[Out]

1/n*a*ln(c*x^n)+1/2/n*b*ln(c*x^n)^2

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maxima [A]  time = 0.44, size = 20, normalized size = 0.91 \[ \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*log(c*x^n) + a)^2/(b*n)

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mupad [B]  time = 3.41, size = 19, normalized size = 0.86 \[ a\,\ln \relax (x)+\frac {b\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/x,x)

[Out]

a*log(x) + (b*log(c*x^n)^2)/(2*n)

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sympy [A]  time = 8.60, size = 34, normalized size = 1.55 \[ \begin {cases} a \log {\relax (x )} & \text {for}\: b = 0 \\- \left (- a - b \log {\relax (c )}\right ) \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x,x)

[Out]

Piecewise((a*log(x), Eq(b, 0)), (-(-a - b*log(c))*log(x), Eq(n, 0)), ((-a - b*log(c*x**n))**2/(2*b*n), True))

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