Integrand size = 19, antiderivative size = 324 \[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}-\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a}-\frac {\log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a}-\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a}+\frac {\operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{a} \] Output:
ln(-d*x/c)*ln(d*x+c)/a-1/3*ln(-d*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*c-a^(1/3)*d) )*ln(d*x+c)/a-1/3*ln(-d*((-1)^(2/3)*a^(1/3)+b^(1/3)*x)/(b^(1/3)*c-(-1)^(2/ 3)*a^(1/3)*d))*ln(d*x+c)/a-1/3*ln((-1)^(1/3)*d*(a^(1/3)+(-1)^(2/3)*b^(1/3) *x)/(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d))*ln(d*x+c)/a-1/3*polylog(2,b^(1/3)*(d *x+c)/(b^(1/3)*c-a^(1/3)*d))/a-1/3*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c+(- 1)^(1/3)*a^(1/3)*d))/a-1/3*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c-(-1)^(2/3) *a^(1/3)*d))/a+polylog(2,1+d*x/c)/a
Time = 0.10 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.02 \[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}-\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a}-\frac {\log \left (-\frac {(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a}-\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a}+\frac {\operatorname {PolyLog}\left (2,\frac {c+d x}{c}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a} \] Input:
Integrate[Log[c + d*x]/(x*(a + b*x^3)),x]
Output:
(Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/ 3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a) - (Log[-(((-1)^(2/3)*d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/ (3*a) - (Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x])/(3*a) + PolyLog[2, (c + d*x)/c]/a - P olyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)]/(3*a)
Time = 1.05 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2863, 2009}
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.
| \(\displaystyle \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {\log (c+d x)}{a x}-\frac {b x^2 \log (c+d x)}{a \left (a+b x^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a}-\frac {\log (c+d x) \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a}-\frac {\log (c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a}\) |
Input:
Int[Log[c + d*x]/(x*(a + b*x^3)),x]
Output:
(Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/ 3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a) - (Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*a) - (Lo g[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)* a^(1/3)*d)]*Log[c + d*x])/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)* c - a^(1/3)*d)]/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^( 1/3)*a^(1/3)*d)]/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^ (2/3)*a^(1/3)*d)]/(3*a) + PolyLog[2, 1 + (d*x)/c]/a
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.33
| method | result | size |
| derivativedivides | \(\frac {\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{3 a}\) | \(106\) |
| default | \(\frac {\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{3 a}\) | \(106\) |
| risch | \(\frac {\ln \left (-\frac {d x}{c}\right ) \ln \left (d x +c \right )}{a}+\frac {\operatorname {dilog}\left (-\frac {d x}{c}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{3 a}\) | \(108\) |
| parts | \(\frac {\ln \left (d x +c \right ) \ln \left (x \right )}{a}-\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{3}+a \right )}{3 a}-\frac {d \left (\frac {3 \operatorname {dilog}\left (\frac {d x +c}{c}\right )}{a d}+\frac {3 \ln \left (x \right ) \ln \left (\frac {d x +c}{c}\right )}{a d}-\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{3}+a \right )}{a d}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{a d}\right )}{3}\) | \(175\) |
Input:
int(ln(d*x+c)/x/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
(dilog(-d*x/c)+ln(d*x+c)*ln(-d*x/c))/a-1/3*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_ R1)+dilog((-d*x+_R1-c)/_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3- b*c^3))/a
\[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x} \,d x } \] Input:
integrate(log(d*x+c)/x/(b*x^3+a),x, algorithm="fricas")
Output:
integral(log(d*x + c)/(b*x^4 + a*x), x)
Timed out. \[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(ln(d*x+c)/x/(b*x**3+a),x)
Output:
Timed out
\[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x} \,d x } \] Input:
integrate(log(d*x+c)/x/(b*x^3+a),x, algorithm="maxima")
Output:
integrate(log(d*x + c)/((b*x^3 + a)*x), x)
\[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x} \,d x } \] Input:
integrate(log(d*x+c)/x/(b*x^3+a),x, algorithm="giac")
Output:
integrate(log(d*x + c)/((b*x^3 + a)*x), x)
Timed out. \[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x\,\left (b\,x^3+a\right )} \,d x \] Input:
int(log(c + d*x)/(x*(a + b*x^3)),x)
Output:
int(log(c + d*x)/(x*(a + b*x^3)), x)
\[ \int \frac {\log (c+d x)}{x \left (a+b x^3\right )} \, dx=\int \frac {\mathrm {log}\left (d x +c \right )}{b \,x^{4}+a x}d x \] Input:
int(log(d*x+c)/x/(b*x^3+a),x)
Output:
int(log(c + d*x)/(a*x + b*x**4),x)