Integrand size = 23, antiderivative size = 54 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d r^2} \] Output:
-(a+b*ln(c*x^n))*ln(1+d/e/(x^r))/d/r+b*n*polylog(2,-d/e/(x^r))/d/r^2
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\frac {b n r^2 \log ^2(x)-2 r \left (a+b \log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b n r \log (x) \left (\log \left (d-d x^r\right )-\log \left (d+e x^r\right )\right )+2 b n \log \left (-\frac {e x^r}{d}\right ) \log \left (d+e x^r\right )+2 b n \operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )}{2 d r^2} \] Input:
Integrate[(a + b*Log[c*x^n])/(x*(d + e*x^r)),x]
Output:
(b*n*r^2*Log[x]^2 - 2*r*(a + b*Log[c*x^n])*Log[d - d*x^r] + 2*b*n*r*Log[x] *(Log[d - d*x^r] - Log[d + e*x^r]) + 2*b*n*Log[-((e*x^r)/d)]*Log[d + e*x^r ] + 2*b*n*PolyLog[2, 1 + (e*x^r)/d])/(2*d*r^2)
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2779, 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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {b n \int \frac {\log \left (\frac {d x^{-r}}{e}+1\right )}{x}dx}{d r}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r}\) |
Input:
Int[(a + b*Log[c*x^n])/(x*(d + e*x^r)),x]
Output:
-(((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d*r)) + (b*n*PolyLog[2, -(d/(e* x^r))])/(d*r^2)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.64 (sec) , antiderivative size = 246, normalized size of antiderivative = 4.56
method | result | size |
risch | \(\frac {b \ln \left (d +e \,x^{r}\right ) n \ln \left (x \right )}{r d}-\frac {b \ln \left (d +e \,x^{r}\right ) \ln \left (x^{n}\right )}{r d}-\frac {b \ln \left (x^{r}\right ) n \ln \left (x \right )}{r d}+\frac {b \ln \left (x^{r}\right ) \ln \left (x^{n}\right )}{r d}+\frac {b n \ln \left (x \right )^{2}}{2 d}-\frac {b n \ln \left (x \right ) \ln \left (1+\frac {e \,x^{r}}{d}\right )}{r d}-\frac {b n \operatorname {polylog}\left (2, -\frac {e \,x^{r}}{d}\right )}{r^{2} d}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d +e \,x^{r}\right )}{r d}+\frac {\ln \left (x^{r}\right )}{r d}\right )\) | \(246\) |
Input:
int((a+b*ln(c*x^n))/x/(d+e*x^r),x,method=_RETURNVERBOSE)
Output:
b/r/d*ln(d+e*x^r)*n*ln(x)-b/r/d*ln(d+e*x^r)*ln(x^n)-b/r/d*ln(x^r)*n*ln(x)+ b/r/d*ln(x^r)*ln(x^n)+1/2*b*n/d*ln(x)^2-b/r*n/d*ln(x)*ln(1+e*x^r/d)-b/r^2* n/d*polylog(2,-e*x^r/d)+(1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b *csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*csgn(I*c*x^n)^3+1/2*I*Pi*b *csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)*(-1/r/d*ln(d+e*x^r)+1/r/d*ln(x^r))
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.72 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\frac {b n r^{2} \log \left (x\right )^{2} - 2 \, b n r \log \left (x\right ) \log \left (\frac {e x^{r} + d}{d}\right ) - 2 \, b n {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - 2 \, {\left (b r \log \left (c\right ) + a r\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b r^{2} \log \left (c\right ) + a r^{2}\right )} \log \left (x\right )}{2 \, d r^{2}} \] Input:
integrate((a+b*log(c*x^n))/x/(d+e*x^r),x, algorithm="fricas")
Output:
1/2*(b*n*r^2*log(x)^2 - 2*b*n*r*log(x)*log((e*x^r + d)/d) - 2*b*n*dilog(-( e*x^r + d)/d + 1) - 2*(b*r*log(c) + a*r)*log(e*x^r + d) + 2*(b*r^2*log(c) + a*r^2)*log(x))/(d*r^2)
Time = 148.83 (sec) , antiderivative size = 452, normalized size of antiderivative = 8.37 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\text {Too large to display} \] Input:
integrate((a+b*ln(c*x**n))/x/(d+e*x**r),x)
Output:
-2*a*e*Piecewise((1/(2*e) + x**r/d, Eq(e, 0)), (-log(-2*e*x**r)/(2*e), Tru e))/(d*r) - 2*a*e*Piecewise((1/(2*e) + x**r/d, Eq(e, 0)), (log(2*d + 2*e*x **r)/(2*e), True))/(d*r) + 2*b*e*n*Piecewise((Piecewise((log(x)/(2*e) + x* *r/(d*r), Ne(r, 0)), (log(x)/(2*e) + log(x)/d, True)), Eq(e, 0)), (-Piecew ise((0, (Abs(e*x**r) < 1) & (1/Abs(e*x**r) < 1)), (log(e*x**r)**2/(2*r) + I*pi*log(e*x**r)/r, Abs(e*x**r) < 1), (log(1/(e*x**r))**2/(2*r) - I*pi*log (1/(e*x**r))/r, 1/Abs(e*x**r) < 1), (meijerg(((), (1, 1, 1)), ((0, 0, 0), ()), e*x**r*exp_polar(I*pi))/r + meijerg(((1, 1, 1), ()), ((), (0, 0, 0)), e*x**r*exp_polar(I*pi))/r, True))/(2*e) - log(2)*log(x)/(2*e), True))/(d* r) + 2*b*e*n*Piecewise((Piecewise((log(x)/(2*e) + x**r/(d*r), Ne(r, 0)), ( log(x)/(2*e) + log(x)/d, True)), Eq(e, 0)), (Piecewise((-polylog(2, e*x**r *exp_polar(I*pi)/d)/r, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - po lylog(2, e*x**r*exp_polar(I*pi)/d)/r, Abs(x) < 1), (-log(d)*log(1/x) - pol ylog(2, e*x**r*exp_polar(I*pi)/d)/r, 1/Abs(x) < 1), (-meijerg(((), (1, 1)) , ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x**r*exp_polar(I*pi)/d)/r, True))/(2*e) + log(2)*log(x)/(2* e), True))/(d*r) - 2*b*e*Piecewise((1/(2*e) + x**r/d, Eq(e, 0)), (-log(-2* e*x**r)/(2*e), True))*log(c*x**n)/(d*r) - 2*b*e*Piecewise((1/(2*e) + x**r/ d, Eq(e, 0)), (log(2*d + 2*e*x**r)/(2*e), True))*log(c*x**n)/(d*r)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x} \,d x } \] Input:
integrate((a+b*log(c*x^n))/x/(d+e*x^r),x, algorithm="maxima")
Output:
a*(log(x)/d - log((e*x^r + d)/e)/(d*r)) + b*integrate((log(c) + log(x^n))/ (e*x*x^r + d*x), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x} \,d x } \] Input:
integrate((a+b*log(c*x^n))/x/(d+e*x^r),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)/((e*x^r + d)*x), x)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+e\,x^r\right )} \,d x \] Input:
int((a + b*log(c*x^n))/(x*(d + e*x^r)),x)
Output:
int((a + b*log(c*x^n))/(x*(d + e*x^r)), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\frac {\left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{x^{r} e x +d x}d x \right ) b d r -\mathrm {log}\left (x^{r} e +d \right ) a +\mathrm {log}\left (x \right ) a r}{d r} \] Input:
int((a+b*log(c*x^n))/x/(d+e*x^r),x)
Output:
(int(log(x**n*c)/(x**r*e*x + d*x),x)*b*d*r - log(x**r*e + d)*a + log(x)*a* r)/(d*r)