3.250 \(\int \frac {1}{a x+b x \log (c x^n)} \, dx\)

Optimal. Leaf size=18 \[ \frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {31} \[ \frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x*Log[c*x^n])^(-1),x]

[Out]

Log[a + b*Log[c*x^n]]/(b*n)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {align*} \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \[ \frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x*Log[c*x^n])^(-1),x]

[Out]

Log[a + b*Log[c*x^n]]/(b*n)

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fricas [A]  time = 0.44, size = 19, normalized size = 1.06 \[ \frac {\log \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*log(c*x^n)),x, algorithm="fricas")

[Out]

log(b*n*log(x) + b*log(c) + a)/(b*n)

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giac [B]  time = 0.16, size = 45, normalized size = 2.50 \[ \frac {\log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi b {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + a\right )}^{2}\right )}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*log(c*x^n)),x, algorithm="giac")

[Out]

1/2*log(1/4*(pi*b*n*(sgn(x) - 1) + pi*b*(sgn(c) - 1))^2 + (b*n*log(abs(x)) + b*log(abs(c)) + a)^2)/(b*n)

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maple [A]  time = 0.07, size = 19, normalized size = 1.06 \[ \frac {\ln \left (b \ln \left (c \,x^{n}\right )+a \right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.66, size = 24, normalized size = 1.33 \[ \frac {\log \left (\frac {b \log \relax (c) + b \log \left (x^{n}\right ) + a}{b}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*log(c*x^n)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.38, size = 18, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b*x*log(c*x^n)),x)

[Out]

log(a + b*log(c*x^n))/(b*n)

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sympy [A]  time = 1.15, size = 34, normalized size = 1.89 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b \log {\relax (c )}} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x*ln(c*x**n)),x)

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0)), (log(x)/(a + b*log(c)), Eq(n, 0)), (log(x)/a, Eq(b, 0)), (log(a/b +
n*log(x) + log(c))/(b*n), True))

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