3.5.20 \(\int \frac {a+b \log (c x^n)}{x (c-x^{-n})} \, dx\) [420]

Optimal. Leaf size=37 \[ \frac {a \log \left (1-c x^n\right )}{c n}-\frac {b \text {Li}_2\left (1-c x^n\right )}{c n} \]

[Out]

a*ln(1-c*x^n)/c/n-b*polylog(2,1-c*x^n)/c/n

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Rubi [A]
time = 0.10, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2378, 2370, 2353, 2352} \begin {gather*} \frac {a \log \left (1-c x^n\right )}{c n}-\frac {b \text {PolyLog}\left (2,1-c x^n\right )}{c n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x*(c - x^(-n))),x]

[Out]

(a*Log[1 - c*x^n])/(c*n) - (b*PolyLog[2, 1 - c*x^n])/(c*n)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(a + b*Log[(-c)*(d/e)])*(Log[d + e*
x]/e), x] + Dist[b, Int[Log[(-e)*(x/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[(-c)*(d/e), 0]

Rule 2370

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2378

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Dist[1/n, Subst[Int[(a
+ b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{\left (c-\frac {1}{x}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{-1+c x} \, dx,x,x^n\right )}{n}\\ &=\frac {a \log \left (1-c x^n\right )}{c n}+\frac {b \text {Subst}\left (\int \frac {\log (c x)}{-1+c x} \, dx,x,x^n\right )}{n}\\ &=\frac {a \log \left (1-c x^n\right )}{c n}-\frac {b \text {Li}_2\left (1-c x^n\right )}{c n}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-c x^n\right )+b \text {Li}_2\left (c x^n\right )}{c n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*x^n])*Log[1 - c*x^n] + b*PolyLog[2, c*x^n])/(c*n)

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Maple [A]
time = 0.18, size = 31, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {a \ln \left (c \,x^{n}-1\right )}{c}-\frac {b \dilog \left (c \,x^{n}\right )}{c}}{n}\)	\(31\)
default	\(\frac {\frac {a \ln \left (c \,x^{n}-1\right )}{c}-\frac {b \dilog \left (c \,x^{n}\right )}{c}}{n}\)	\(31\)
risch	\(\frac {b \ln \left (1-c \,x^{n}\right ) \ln \left (x^{n}\right )}{n c}-\frac {b \ln \left (1-c \,x^{n}\right ) \ln \left (c \,x^{n}\right )}{n c}-\frac {b \dilog \left (c \,x^{n}\right )}{n c}-\frac {i \ln \left (c \,x^{n}-1\right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n c}+\frac {i \ln \left (c \,x^{n}-1\right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 n c}+\frac {i \ln \left (c \,x^{n}-1\right ) b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 n c}-\frac {i \ln \left (c \,x^{n}-1\right ) b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 n c}+\frac {\ln \left (c \,x^{n}-1\right ) b \ln \left (c \right )}{n c}+\frac {\ln \left (c \,x^{n}-1\right ) a}{n c}\)	\(234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x/(c-1/(x^n)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate((x^n*log(c) + x^n*log(x^n))/(c*x*x^n - x), x) + a*log((c*x^n - 1)/c)/(c*n)

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Fricas [A]
time = 0.37, size = 45, normalized size = 1.22 \begin {gather*} \frac {b n \log \left (-c x^{n} + 1\right ) \log \left (x\right ) + b {\rm Li}_2\left (c x^{n}\right ) + {\left (b \log \left (c\right ) + a\right )} \log \left (c x^{n} - 1\right )}{c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)/((c - 1/x^n)*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (c-\frac {1}{x^n}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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