Integrand size = 16, antiderivative size = 247 \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}} \]
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Time = 0.25 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2456, 2436, 2332, 2441, 2440, 2438} \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {x}{c} \]
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Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c \left (d+c x^2\right )}\right ) \, dx \\ & = \frac {\int \log (a+b x) \, dx}{c}-\frac {d \int \frac {\log (a+b x)}{d+c x^2} \, dx}{c} \\ & = \frac {\text {Subst}(\int \log (x) \, dx,x,a+b x)}{b c}-\frac {d \int \left (\frac {\log (a+b x)}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\log (a+b x)}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{c} \\ & = -\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \int \frac {\log (a+b x)}{\sqrt {d}-\sqrt {-c} x} \, dx}{2 c}-\frac {\sqrt {d} \int \frac {\log (a+b x)}{\sqrt {d}+\sqrt {-c} x} \, dx}{2 c} \\ & = -\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 {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{a+b x} \, dx}{2 (-c)^{3/2}}-\frac {\left (b \sqrt {d}\right ) \int \frac {\log \left (\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-a \sqrt {-c}+b \sqrt {d}}\right )}{a+b x} \, dx}{2 (-c)^{3/2}} \\ & = -\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} \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c} x}{-a \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,a+b x\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c} x}{a \sqrt {-c}+b \sqrt {d}}\right )}{x} \, dx,x,a+b x\right )}{2 (-c)^{3/2}} \\ & = -\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} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00 \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=-\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}} \]
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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 | \(\frac {\ln \left (b x +a \right ) x}{c}+\frac {\ln \left (b x +a \right ) a}{b c}-\frac {x}{c}-\frac {a}{b c}-\frac {d \ln \left (b x +a \right ) \ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )}{2 c \sqrt {-c d}}+\frac {d \ln \left (b x +a \right ) \ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 c \sqrt {-c d}}-\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )}{2 c \sqrt {-c d}}+\frac {d \operatorname {dilog}\left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 c \sqrt {-c d}}\) | \(248\) |
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\[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\log \left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \]
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\[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {x^{2} \log {\left (a + b x \right )}}{c x^{2} + d}\, dx \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.21 \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=-{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \log \left (b x + a\right ) - \frac {2 \, b c x - 2 \, a c \log \left (b x + a\right ) + {\left (b \arctan \left (\frac {{\left (b^{2} x + a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + b^{2} d}, \frac {a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) \log \left (c x^{2} + d\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) + i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d + {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right ) - i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d - {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right )\right )} \sqrt {c} \sqrt {d}}{2 \, b c^{2}} \]
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\[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\log \left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {\ln \left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \]
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