Integrand size = 20, antiderivative size = 227 \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {2 \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{d} \]
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Time = 0.25 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2515, 2512, 266, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {\log \left (\frac {a}{x^2}+b\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} d+\sqrt {b} c}\right )}{d}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}+\frac {2 \operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]
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Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2512
Rule 2515
Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (b+\frac {a}{x^2}\right )}{c+d x} \, dx \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {(2 a) \int \frac {\log (c+d x)}{\left (b+\frac {a}{x^2}\right ) x^3} \, dx}{d} \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {(2 a) \int \left (\frac {\log (c+d x)}{a x}-\frac {b x \log (c+d x)}{a \left (a+b x^2\right )}\right ) \, dx}{d} \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \int \frac {\log (c+d x)}{x} \, dx}{d}-\frac {(2 b) \int \frac {x \log (c+d x)}{a+b x^2} \, dx}{d} \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-2 \int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx-\frac {(2 b) \int \left (-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{d} \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\frac {\sqrt {b} \int \frac {\log (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{d}-\frac {\sqrt {b} \int \frac {\log (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{d} \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\int \frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right )}{c+d x} \, dx+\int \frac {\log \left (\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {-a} d}\right )}{c+d x} \, dx \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {-a} d}\right )}{x} \, dx,x,c+d x\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {-a} d}\right )}{x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\text {Li}_2\left (\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d}+\frac {2 \text {Li}_2\left (1+\frac {d x}{c}\right )}{d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\frac {\log \left (b+\frac {a}{x^2}\right ) \log (c+d x)}{d}+\frac {2 \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (\frac {d \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-a} d}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} c-\sqrt {-a} d}\right ) \log (c+d x)}{d}+\frac {2 \operatorname {PolyLog}\left (2,\frac {c+d x}{c}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{d} \]
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Time = 1.69 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.01
method | result | size |
parts | \(\frac {\ln \left (\frac {b \,x^{2}+a}{x^{2}}\right ) \ln \left (d x +c \right )}{d}-\frac {2 \left (-b \,d^{3} \left (-\frac {\ln \left (d x +c \right ) \left (\ln \left (\frac {d \sqrt {-a b}+c b -b \left (d x +c \right )}{d \sqrt {-a b}+c b}\right )+\ln \left (\frac {d \sqrt {-a b}-c b +b \left (d x +c \right )}{d \sqrt {-a b}-c b}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {-a b}+c b -b \left (d x +c \right )}{d \sqrt {-a b}+c b}\right )+\operatorname {dilog}\left (\frac {d \sqrt {-a b}-c b +b \left (d x +c \right )}{d \sqrt {-a b}-c b}\right )}{2 b}\right )-d^{3} \left (\operatorname {dilog}\left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )\right )\right )}{d^{4}}\) | \(230\) |
derivativedivides | \(-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )-2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{d}+\frac {\left (\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (b +\frac {a}{x^{2}}\right )}{c}-\frac {2 a \left (\frac {\ln \left (\frac {c}{x}+d \right ) \left (\ln \left (\frac {c \sqrt {-a b}+a d -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+a d}\right )+\ln \left (\frac {c \sqrt {-a b}-a d +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-a d}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}+a d -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-a b}-a d +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-a d}\right )}{2 a}\right )}{c}\right ) c}{d}\) | \(330\) |
default | \(-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )-2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{d}+\frac {\left (\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (b +\frac {a}{x^{2}}\right )}{c}-\frac {2 a \left (\frac {\ln \left (\frac {c}{x}+d \right ) \left (\ln \left (\frac {c \sqrt {-a b}+a d -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+a d}\right )+\ln \left (\frac {c \sqrt {-a b}-a d +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-a d}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}+a d -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {c \sqrt {-a b}-a d +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-a d}\right )}{2 a}\right )}{c}\right ) c}{d}\) | \(330\) |
risch | \(\frac {\ln \left (\frac {1}{x}\right ) \ln \left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\ln \left (\frac {1}{x}\right ) \ln \left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {\ln \left (\frac {1}{x}\right ) \ln \left (b +\frac {a}{x^{2}}\right )}{d}+\frac {\operatorname {dilog}\left (\frac {-\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\operatorname {dilog}\left (\frac {\frac {a}{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (b +\frac {a}{x^{2}}\right )}{d}-\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (\frac {c \sqrt {-a b}+a d -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+a d}\right )}{d}-\frac {\ln \left (\frac {c}{x}+d \right ) \ln \left (\frac {c \sqrt {-a b}-a d +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-a d}\right )}{d}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}+a d -a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}+a d}\right )}{d}-\frac {\operatorname {dilog}\left (\frac {c \sqrt {-a b}-a d +a \left (\frac {c}{x}+d \right )}{c \sqrt {-a b}-a d}\right )}{d}\) | \(335\) |
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\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c} \,d x } \]
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\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int \frac {\log {\left (\frac {a}{x^{2}} + b \right )}}{c + d x}\, dx \]
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\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c} \,d x } \]
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\[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int { \frac {\log \left (\frac {b x^{2} + a}{x^{2}}\right )}{d x + c} \,d x } \]
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Timed out. \[ \int \frac {\log \left (\frac {a+b x^2}{x^2}\right )}{c+d x} \, dx=\int \frac {\ln \left (\frac {b\,x^2+a}{x^2}\right )}{c+d\,x} \,d x \]
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