Integrand size = 23, antiderivative size = 357 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 p}{d x}+\frac {2 \sqrt {a} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^2}+\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )}{2 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^2}+\frac {2 e p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d^2} \]
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Time = 0.33 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2516, 2505, 269, 331, 211, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {a} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}+\frac {e \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e p \operatorname {PolyLog}\left (2,\frac {b}{a x^2}+1\right )}{2 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^2}+\frac {2 e p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}+\frac {2 p}{d x} \]
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Rule 211
Rule 266
Rule 269
Rule 331
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2504
Rule 2505
Rule 2512
Rule 2516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x^2}-\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx}{d^2} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x^2}\right )}{2 d^2}-\frac {(2 b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^4} \, dx}{d}+\frac {(2 b e p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{d^2} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}-\frac {(2 b p) \int \frac {1}{x^2 \left (b+a x^2\right )} \, dx}{d}-\frac {(b e p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x^2}\right )}{2 d^2}+\frac {(2 b e p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a x \log (d+e x)}{b \left (b+a x^2\right )}\right ) \, dx}{d^2} \\ & = \frac {2 p}{d x}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^2}+\frac {(2 a p) \int \frac {1}{b+a x^2} \, dx}{d}+\frac {(2 e p) \int \frac {\log (d+e x)}{x} \, dx}{d^2}-\frac {(2 a e p) \int \frac {x \log (d+e x)}{b+a x^2} \, dx}{d^2} \\ & = \frac {2 p}{d x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^2}-\frac {(2 a e p) \int \left (-\frac {\sqrt {-a} \log (d+e x)}{2 a \left (\sqrt {b}-\sqrt {-a} x\right )}+\frac {\sqrt {-a} \log (d+e x)}{2 a \left (\sqrt {b}+\sqrt {-a} x\right )}\right ) \, dx}{d^2}-\frac {\left (2 e^2 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d^2} \\ & = \frac {2 p}{d x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^2}+\frac {2 e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2}+\frac {\left (\sqrt {-a} e p\right ) \int \frac {\log (d+e x)}{\sqrt {b}-\sqrt {-a} x} \, dx}{d^2}-\frac {\left (\sqrt {-a} e p\right ) \int \frac {\log (d+e x)}{\sqrt {b}+\sqrt {-a} x} \, dx}{d^2} \\ & = \frac {2 p}{d x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^2}+\frac {2 e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2}+\frac {\left (e^2 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d+e x} \, dx}{d^2}+\frac {\left (e^2 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right )}{d+e x} \, dx}{d^2} \\ & = \frac {2 p}{d x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^2}+\frac {2 e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a} x}{-\sqrt {-a} d+\sqrt {b} e}\right )}{x} \, dx,x,d+e x\right )}{d^2}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a} x}{\sqrt {-a} d+\sqrt {b} e}\right )}{x} \, dx,x,d+e x\right )}{d^2} \\ & = \frac {2 p}{d x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^2}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^2}+\frac {2 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^2}+\frac {e p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^2}+\frac {2 e p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {\frac {4 d p}{x}-\frac {4 \sqrt {a} d p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}-\frac {2 d \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )+2 e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)+4 e p \log \left (-\frac {e x}{d}\right ) \log (d+e x)-2 e p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-2 e p \log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+e p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )-2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )-2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )+4 e p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^2} \]
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Time = 1.34 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.17
method | result | size |
parts | \(\frac {e \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{d x}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) e \ln \left (x \right )}{d^{2}}+2 p b \left (\frac {e \left (-\frac {a \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\ln \left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )}{2 a}\right )}{b}+\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b}\right )}{d^{2}}+\frac {1}{d b x}+\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{d b \sqrt {a b}}-\frac {e \left (\frac {\ln \left (x \right )^{2}}{2 b}-\frac {a \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{b}\right )}{d^{2}}\right )\) | \(418\) |
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
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