Integrand size = 23, antiderivative size = 388 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^3}\right )}{3 d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d}-\frac {3 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d} \]
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Time = 0.42 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 21, 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} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x^3}\right ) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d}+\frac {p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {b}{a x^3}+1\right )}{3 d}-\frac {3 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
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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^3}\right )^p\right )}{d x}-\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\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^3}\right )}{3 d}-\frac {(3 b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\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^3}\right )}{3 d}-\frac {(3 b p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a x^2 \log (d+e x)}{b \left (b+a x^3\right )}\right ) \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d}-\frac {(3 p) \int \frac {\log (d+e x)}{x} \, dx}{d}+\frac {(3 a p) \int \frac {x^2 \log (d+e x)}{b+a x^3} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d}+\frac {(3 a p) \int \left (\frac {\log (d+e x)}{3 a^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac {\log (d+e x)}{3 a^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac {\log (d+e x)}{3 a^{2/3} \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}\right ) \, dx}{d}+\frac {(3 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^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d}-\frac {3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}+\frac {\left (\sqrt [3]{a} p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d}+\frac {\left (\sqrt [3]{a} p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d}+\frac {\left (\sqrt [3]{a} p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d}-\frac {3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{d+e x} \, dx}{d}-\frac {(e p) \int \frac {\log \left (\frac {e \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{d}-\frac {(e p) \int \frac {\log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d}-\frac {3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d}+\frac {p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d}-\frac {3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.02 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d}-\frac {3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x^3}}{a}\right )}{3 d}-\frac {3 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.55 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.57
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (x \right )}{d}+3 p b \left (\frac {\ln \left (x \right )^{2}}{2 d b}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a +b \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )}{3 d b}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a -3 \textit {\_Z}^{2} a d +3 \textit {\_Z} a \,d^{2}-a \,d^{3}+e^{3} b \right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )}{3 d b}-\frac {\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d b}-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )}{d b}\right )\) | \(220\) |
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x^3}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \]
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