Integrand size = 38, antiderivative size = 28 \[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {(1-a c) \left (e+d x^{-m}\right )}{e}\right )}{d m} \] Output:
polylog(2,(-a*c+1)*(e+d/(x^m))/e)/d/m
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {(-1+a c) x^{-m} \left (d+e x^m\right )}{e}\right )}{d m} \] Input:
Integrate[Log[(-d + a*c*d + a*c*e*x^m)/(e*x^m)]/(x*(d + e*x^m)),x]
Output:
PolyLog[2, -(((-1 + a*c)*(d + e*x^m))/(e*x^m))]/(d*m)
Time = 0.68 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2930, 2925, 2005, 2840, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\log \left (\frac {x^{-m} \left (a c d+a c e x^m-d\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx\) |
\(\Big \downarrow \) 2930 |
\(\displaystyle \int \frac {\log \left (\frac {x^{-m} (a c d-d)}{e}+a c\right )}{x \left (d+e x^m\right )}dx\) |
\(\Big \downarrow \) 2925 |
\(\displaystyle -\frac {\int \frac {x^m \log \left (a c-\frac {(1-a c) d x^{-m}}{e}\right )}{e x^m+d}dx^{-m}}{m}\) |
\(\Big \downarrow \) 2005 |
\(\displaystyle -\frac {\int \frac {\log \left (a c-\frac {(1-a c) d x^{-m}}{e}\right )}{d x^{-m}+e}dx^{-m}}{m}\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle -\frac {\int x^m \log \left (1-\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )d\left (d x^{-m}+e\right )}{d m}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\operatorname {PolyLog}\left (2,\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m}\) |
Input:
Int[Log[(-d + a*c*d + a*c*e*x^m)/(e*x^m)]/(x*(d + e*x^m)),x]
Output:
PolyLog[2, ((1 - a*c)*(e + d/x^m))/e]/(d*m)
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ Symbol] :> Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(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] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*(u_)^(r_.)*((h_.)*(x_))^(m_ .), x_Symbol] :> Int[(h*x)^m*ExpandToSum[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]
Time = 6.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (\frac {\left (a c d -d \right ) x^{-m}}{e}+a c \right )}{m d}\) | \(30\) |
default | \(\frac {\operatorname {dilog}\left (\frac {\left (a c d -d \right ) x^{-m}}{e}+a c \right )}{m d}\) | \(30\) |
risch | \(\text {Expression too large to display}\) | \(1200\) |
Input:
int(ln((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x,method=_RETURNVERBOSE)
Output:
1/m*dilog(1/e*(a*c*d-d)/(x^m)+a*c)/d
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\frac {{\rm Li}_2\left (-\frac {a c e x^{m} + {\left (a c - 1\right )} d}{e x^{m}} + 1\right )}{d m} \] Input:
integrate(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x, algorithm="fric as")
Output:
dilog(-(a*c*e*x^m + (a*c - 1)*d)/(e*x^m) + 1)/(d*m)
Exception generated. \[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(ln((-d+a*c*d+a*c*e*x**m)/e/(x**m))/x/(d+e*x**m),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\int { \frac {\log \left (\frac {a c e x^{m} + a c d - d}{e x^{m}}\right )}{{\left (e x^{m} + d\right )} x} \,d x } \] Input:
integrate(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x, algorithm="maxi ma")
Output:
(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)
\[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\int { \frac {\log \left (\frac {a c e x^{m} + a c d - d}{e x^{m}}\right )}{{\left (e x^{m} + d\right )} x} \,d x } \] Input:
integrate(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x, algorithm="giac ")
Output:
integrate(log((a*c*e*x^m + a*c*d - d)/(e*x^m))/((e*x^m + d)*x), x)
Timed out. \[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\int \frac {\ln \left (\frac {a\,c\,d-d+a\,c\,e\,x^m}{e\,x^m}\right )}{x\,\left (d+e\,x^m\right )} \,d x \] Input:
int(log((a*c*d - d + a*c*e*x^m)/(e*x^m))/(x*(d + e*x^m)),x)
Output:
int(log((a*c*d - d + a*c*e*x^m)/(e*x^m))/(x*(d + e*x^m)), x)
\[ \int \frac {\log \left (\frac {x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (\frac {x^{m} a c e +a c d -d}{x^{m} e}\right )}{x^{2 m} a c \,e^{2} x +2 x^{m} a c d e x -x^{m} d e x +a c \,d^{2} x -d^{2} x}d x \right ) a c \,d^{2} m +2 \left (\int \frac {\mathrm {log}\left (\frac {x^{m} a c e +a c d -d}{x^{m} e}\right )}{x^{2 m} a c \,e^{2} x +2 x^{m} a c d e x -x^{m} d e x +a c \,d^{2} x -d^{2} x}d x \right ) d^{2} m -\mathrm {log}\left (\frac {x^{m} a c e +a c d -d}{x^{m} e}\right )^{2} a c}{2 d m \left (a c -1\right )} \] Input:
int(log((-d+a*c*d+a*c*e*x^m)/e/(x^m))/x/(d+e*x^m),x)
Output:
( - 2*int(log((x**m*a*c*e + a*c*d - d)/(x**m*e))/(x**(2*m)*a*c*e**2*x + 2* x**m*a*c*d*e*x - x**m*d*e*x + a*c*d**2*x - d**2*x),x)*a*c*d**2*m + 2*int(l og((x**m*a*c*e + a*c*d - d)/(x**m*e))/(x**(2*m)*a*c*e**2*x + 2*x**m*a*c*d* e*x - x**m*d*e*x + a*c*d**2*x - d**2*x),x)*d**2*m - log((x**m*a*c*e + a*c* d - d)/(x**m*e))**2*a*c)/(2*d*m*(a*c - 1))