3.48 \(\int \frac{a+b \log (c x^n)}{x^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac{a+b \log \left (c x^n\right )}{x}-\frac{b n}{x} \]

[Out]

-((b*n)/x) - (a + b*Log[c*x^n])/x

________________________________________________________________________________________

Rubi [A]  time = 0.0129638, antiderivative size = 23, 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 = {2304} \[ -\frac{a+b \log \left (c x^n\right )}{x}-\frac{b n}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/x^2,x]

[Out]

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

Mathematica [A]  time = 0.001195, size = 26, normalized size = 1.13 \[ -\frac{a}{x}-\frac{b \log \left (c x^n\right )}{x}-\frac{b n}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/x^2,x]

[Out]

-(a/x) - (b*n)/x - (b*Log[c*x^n])/x

________________________________________________________________________________________

Maple [C]  time = 0.082, size = 112, normalized size = 4.9 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{x}}-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +2\,bn+2\,a}{2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/x*ln(x^n)-1/2*(I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*b*Pi*csgn(I*
c*x^n)^3+I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*b*n+2*a)/x

________________________________________________________________________________________

Maxima [A]  time = 1.18859, size = 35, normalized size = 1.52 \begin{align*} -\frac{b n}{x} - \frac{b \log \left (c x^{n}\right )}{x} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2,x, algorithm="maxima")

[Out]

-b*n/x - b*log(c*x^n)/x - a/x

________________________________________________________________________________________

Fricas [A]  time = 0.961065, size = 51, normalized size = 2.22 \begin{align*} -\frac{b n \log \left (x\right ) + b n + b \log \left (c\right ) + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2,x, algorithm="fricas")

[Out]

-(b*n*log(x) + b*n + b*log(c) + a)/x

________________________________________________________________________________________

Sympy [A]  time = 0.473747, size = 24, normalized size = 1.04 \begin{align*} - \frac{a}{x} - \frac{b n \log{\left (x \right )}}{x} - \frac{b n}{x} - \frac{b \log{\left (c \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**2,x)

[Out]

-a/x - b*n*log(x)/x - b*n/x - b*log(c)/x

________________________________________________________________________________________

Giac [A]  time = 1.17494, size = 32, normalized size = 1.39 \begin{align*} -\frac{b n \log \left (x\right )}{x} - \frac{b n + b \log \left (c\right ) + a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^2,x, algorithm="giac")

[Out]

-b*n*log(x)/x - (b*n + b*log(c) + a)/x