Integrand size = 16, antiderivative size = 247 \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}} \]
-x/c+(b*x+a)*ln(b*x+a)/b/c+1/2*ln(b*x+a)*ln(-b*(x*(-c)^(1/2)+d^(1/2))/(a*( -c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/2*ln(b*x+a)*ln(b*(-x*(-c)^(1/2) +d^(1/2))/(a*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/2*polylog(2,(b*x+ a)*(-c)^(1/2)/(a*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/2*polylog(2,( b*x+a)*(-c)^(1/2)/(a*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)
Time = 0.13 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00 \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}} \]
-(x/c) + ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[a + b*x]*Log[(b*(Sq rt[d] - Sqrt[-c]*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]* Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))])/ (2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b* Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sq rt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2))
Time = 0.49 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2856, 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.
| \(\displaystyle \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c \left (c x^2+d\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {x}{c}\) |
-(x/c) + ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[a + b*x]*Log[(b*(Sq rt[d] - Sqrt[-c]*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]* Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))])/ (2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b* Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sq rt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2))
3.4.9.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Time = 0.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{c}-\frac {d \,b^{2} \left (-\frac {\ln \left (b x +a \right ) \left (-\ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )+\ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )\right )}{2 b \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )-\operatorname {dilog}\left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 b \sqrt {-c d}}\right )}{c}}{b}\) | \(220\) |
| default | \(\frac {\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{c}-\frac {d \,b^{2} \left (-\frac {\ln \left (b x +a \right ) \left (-\ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )+\ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )\right )}{2 b \sqrt {-c d}}+\frac {\operatorname {dilog}\left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )-\operatorname {dilog}\left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 b \sqrt {-c d}}\right )}{c}}{b}\) | \(220\) |
| risch | \(\frac {\ln \left (b x +a \right ) x}{c}+\frac {\ln \left (b x +a \right ) a}{b c}-\frac {x}{c}-\frac {a}{b c}-\frac {d \ln \left (b x +a \right ) \ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )}{2 c \sqrt {-c d}}+\frac {d \ln \left (b x +a \right ) \ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 c \sqrt {-c d}}-\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )}{2 c \sqrt {-c d}}+\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 c \sqrt {-c d}}\) | \(248\) |
1/b*(1/c*((b*x+a)*ln(b*x+a)-b*x-a)-d*b^2/c*(-1/2*ln(b*x+a)*(-ln((b*(-c*d)^ (1/2)+c*a-c*(b*x+a))/(b*(-c*d)^(1/2)+c*a))+ln((b*(-c*d)^(1/2)-c*a+c*(b*x+a ))/(b*(-c*d)^(1/2)-c*a)))/b/(-c*d)^(1/2)+1/2*(dilog((b*(-c*d)^(1/2)+c*a-c* (b*x+a))/(b*(-c*d)^(1/2)+c*a))-dilog((b*(-c*d)^(1/2)-c*a+c*(b*x+a))/(b*(-c *d)^(1/2)-c*a)))/b/(-c*d)^(1/2)))
\[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\log \left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \]
\[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {x^{2} \log {\left (a + b x \right )}}{c x^{2} + d}\, dx \]
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.21 \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=-{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \log \left (b x + a\right ) - \frac {2 \, b c x - 2 \, a c \log \left (b x + a\right ) + {\left (b \arctan \left (\frac {{\left (b^{2} x + a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + b^{2} d}, \frac {a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) \log \left (c x^{2} + d\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) + i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d + {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right ) - i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d - {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right )\right )} \sqrt {c} \sqrt {d}}{2 \, b c^{2}} \]
-(d*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c) - x/c)*log(b*x + a) - 1/2*(2*b*c*x - 2*a*c*log(b*x + a) + (b*arctan2((b^2*x + a*b)*sqrt(c)*sqrt(d)/(a^2*c + b^2*d), (a*b*c*x + a^2*c)/(a^2*c + b^2*d))*log(c*x^2 + d) - b*arctan(sqrt( c)*x/sqrt(d))*log((b^2*c*x^2 + 2*a*b*c*x + a^2*c)/(a^2*c + b^2*d)) + I*b*d ilog(-(a*b*c*x + b^2*d + (I*b^2*x - I*a*b)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a *b*sqrt(c)*sqrt(d) - b^2*d)) - I*b*dilog(-(a*b*c*x + b^2*d - (I*b^2*x - I* a*b)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*b*sqrt(c)*sqrt(d) - b^2*d)))*sqrt(c)* sqrt(d))/(b*c^2)
\[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\log \left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \]
Timed out. \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {\ln \left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \]