Integrand size = 23, antiderivative size = 287 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )}{2 d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}-\frac {2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d} \]
[Out]
Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2516, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {b}{a x^2}+1\right )}{2 d}-\frac {2 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
[In]
[Out]
Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2504
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}-\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {\text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x^2}\right )}{2 d}-\frac {(2 b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}+\frac {(b 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}-\frac {(2 b 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} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {(2 p) \int \frac {\log (d+e x)}{x} \, dx}{d}+\frac {(2 a p) \int \frac {x \log (d+e x)}{b+a x^2} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}+\frac {(2 a 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}+\frac {(2 e p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {\left (\sqrt {-a} p\right ) \int \frac {\log (d+e x)}{\sqrt {b}-\sqrt {-a} x} \, dx}{d}+\frac {\left (\sqrt {-a} p\right ) \int \frac {\log (d+e x)}{\sqrt {b}+\sqrt {-a} x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {(e p) \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}-\frac {(e p) \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} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {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}-\frac {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} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d}+\frac {p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d}-\frac {2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac {2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x^2}}{a}\right )}{2 d}-\frac {2 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d} \]
[In]
[Out]
Time = 1.07 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.26
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (x \right )}{d}+2 p b \left (\frac {\frac {\ln \left (x \right )^{2}}{2 b}-\frac {a \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{b}}{d}-\frac {-\frac {a \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\ln \left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )}{2 a}\right )}{b}+\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b}}{d}\right )\) | \(362\) |
[In]
[Out]
\[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
[In]
[Out]
\[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \]
[In]
[Out]