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]

(a + b*Log[c*x^n])^2/(2*b*n)

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Rubi [A]  time = 0.0122896, antiderivative size = 22, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0010606, 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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Maple [A]  time = 0.035, size = 27, normalized size = 1.2 \begin{align*}{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}b}{2\,n}}+{\frac{a\ln \left ( c{x}^{n} \right ) }{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11712, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{2 \, b n} \end{align*}

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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Fricas [A]  time = 0.87814, size = 57, normalized size = 2.59 \begin{align*} \frac{1}{2} \, b n \log \left (x\right )^{2} +{\left (b \log \left (c\right ) + a\right )} \log \left (x\right ) \end{align*}

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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Sympy [A]  time = 6.94959, size = 36, normalized size = 1.64 \begin{align*} - \begin{cases} - a \log{\left (x \right )} & \text{for}\: b = 0 \\- \left (a + b \log{\left (c \right )}\right ) \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text{otherwise} \end{cases} \end{align*}

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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Giac [A]  time = 1.13865, size = 26, normalized size = 1.18 \begin{align*} \frac{1}{2} \, b n \log \left (x\right )^{2} + b \log \left (c\right ) \log \left (x\right ) + a \log \left (x\right ) \end{align*}

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)