3.336 \(\int \frac {\log (\frac {x^{-m} (-d+a c d+a c e x^m)}{e})}{x (d+e x^m)} \, dx\)

Optimal. Leaf size=28 \[ \frac {\text {Li}_2\left (\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]

[Out]

polylog(2,(-a*c+1)*(e+d/(x^m))/e)/d/m

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Rubi [A]  time = 0.18, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2480, 2475, 2412, 2393, 2391} \[ \frac {\text {PolyLog}\left (2,\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]

Antiderivative was successfully verified.

[In]

Int[Log[(-d + a*c*d + a*c*e*x^m)/(e*x^m)]/(x*(d + e*x^m)),x]

[Out]

PolyLog[2, ((1 - a*c)*(e + d/x^m))/e]/(d*m)

Rule 2391

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2412

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

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 2480

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*(u_)^(r_.)*((h_.)*(x_))^(m_.), x_Symbol] :> Int[(h*x)^m*Expand
ToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && BinomialQ[{u,
v}, x] &&  !BinomialMatchQ[{u, v}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.11 \[ \frac {\text {Li}_2\left (-\frac {(a c-1) x^{-m} \left (e x^m+d\right )}{e}\right )}{d m} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(-d + a*c*d + a*c*e*x^m)/(e*x^m)]/(x*(d + e*x^m)),x]

[Out]

PolyLog[2, -(((-1 + a*c)*(d + e*x^m))/(e*x^m))]/(d*m)

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fricas [A]  time = 0.43, size = 35, normalized size = 1.25 \[ \frac {{\rm Li}_2\left (-\frac {a c e x^{m} + {\left (a c - 1\right )} d}{e x^{m}} + 1\right )}{d m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x, algorithm="fricas")

[Out]

dilog(-(a*c*e*x^m + (a*c - 1)*d)/(e*x^m) + 1)/(d*m)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {a c e x^{m} + a c d - d}{e x^{m}}\right )}{{\left (e x^{m} + d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x, algorithm="giac")

[Out]

integrate(log((a*c*e*x^m + a*c*d - d)/(e*x^m))/((e*x^m + d)*x), x)

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maple [A]  time = 0.05, size = 28, normalized size = 1.00 \[ \frac {\dilog \left (a c +\frac {\left (a c -1\right ) d \,x^{-m}}{e}\right )}{d m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(e*x^m+d),x)

[Out]

1/m*dilog(a*c+d*(a*c-1)/e/(x^m))/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (a c m - m\right )} \int \frac {\log \relax (x)}{a c e x x^{m} + {\left (a c d - d\right )} x}\,{d x} + \frac {\log \left (a c e x^{m} + {\left (a c - 1\right )} d\right ) \log \relax (x) - \log \relax (e) \log \relax (x) - \log \relax (x) \log \left (x^{m}\right )}{d} + \frac {\log \relax (e) \log \left (\frac {e x^{m} + d}{e}\right )}{d m} + \frac {\log \left (x^{m}\right ) \log \left (\frac {e x^{m}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e x^{m}}{d}\right )}{d m} - \frac {\log \left (a c e x^{m} + {\left (a c - 1\right )} d\right ) \log \left (\frac {a c e x^{m} + a c d - d}{d} + 1\right ) + {\rm Li}_2\left (-\frac {a c e x^{m} + a c d - d}{d}\right )}{d m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x, algorithm="maxima")

[Out]

(a*c*m - m)*integrate(log(x)/(a*c*e*x*x^m + (a*c*d - d)*x), x) + (log(a*c*e*x^m + (a*c - 1)*d)*log(x) - log(e)
*log(x) - log(x)*log(x^m))/d + log(e)*log((e*x^m + d)/e)/(d*m) + (log(x^m)*log(e*x^m/d + 1) + dilog(-e*x^m/d))
/(d*m) - (log(a*c*e*x^m + (a*c - 1)*d)*log((a*c*e*x^m + a*c*d - d)/d + 1) + dilog(-(a*c*e*x^m + a*c*d - d)/d))
/(d*m)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (\frac {a\,c\,d-d+a\,c\,e\,x^m}{e\,x^m}\right )}{x\,\left (d+e\,x^m\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((a*c*d - d + a*c*e*x^m)/(e*x^m))/(x*(d + e*x^m)),x)

[Out]

int(log((a*c*d - d + a*c*e*x^m)/(e*x^m))/(x*(d + e*x^m)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln((-d+a*c*d+a*c*e*x**m)/e/(x**m))/x/(d+e*x**m),x)

[Out]

Exception raised: HeuristicGCDFailed

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