Integrand size = 23, antiderivative size = 737 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\sqrt {3} a^{2/3} p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{a} e p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}+\frac {a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac {\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}-\frac {a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac {\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^3}\right )}{3 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}-\frac {3 e^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d^3} \]
[Out]
Time = 0.62 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {2516, 2505, 269, 331, 206, 31, 648, 631, 210, 642, 298, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {\sqrt {3} a^{2/3} p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac {\sqrt [3]{a} e p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{b} d^2}-\frac {a^{2/3} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 b^{2/3} d}+\frac {a^{2/3} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{a} e p \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac {e^2 \log \left (-\frac {b}{a x^3}\right ) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}-\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b}{a x^3}+1\right )}{3 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {\sqrt [3]{a} e p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} d^2}-\frac {3 e^2 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {3 e p}{d^2 x}+\frac {3 p}{4 d x^2} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 266
Rule 269
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
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^3}\right )^p\right )}{d x^3}-\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x^2}+\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^3 x}-\frac {e^3 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^3 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx}{d^3} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {e^2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x^3}\right )}{3 d^3}-\frac {(3 b p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^6} \, dx}{2 d}+\frac {(3 b e p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^5} \, dx}{d^2}-\frac {\left (3 b e^2 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx}{d^3} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {(3 b p) \int \frac {1}{x^3 \left (b+a x^3\right )} \, dx}{2 d}+\frac {(3 b e p) \int \frac {1}{x^2 \left (b+a x^3\right )} \, dx}{d^2}+\frac {\left (b e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x^3}\right )}{3 d^3}-\frac {\left (3 b e^2 p\right ) \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^3} \\ & = \frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d^3}+\frac {(3 a p) \int \frac {1}{b+a x^3} \, dx}{2 d}-\frac {(3 a e p) \int \frac {x}{b+a x^3} \, dx}{d^2}-\frac {\left (3 e^2 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{d^3}+\frac {\left (3 a e^2 p\right ) \int \frac {x^2 \log (d+e x)}{b+a x^3} \, dx}{d^3} \\ & = \frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d^3}+\frac {(a p) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{2 b^{2/3} d}+\frac {(a p) \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 b^{2/3} d}+\frac {\left (a^{2/3} e p\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{\sqrt [3]{b} d^2}-\frac {\left (a^{2/3} e p\right ) \int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{\sqrt [3]{b} d^2}+\frac {\left (3 a e^2 p\right ) \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^3}+\frac {\left (3 e^3 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d^3} \\ & = \frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}+\frac {a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac {\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d^3}-\frac {3 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}-\frac {\left (a^{2/3} p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 b^{2/3} d}+\frac {(3 a p) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 \sqrt [3]{b} d}-\frac {\left (3 a^{2/3} e p\right ) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 d^2}-\frac {\left (\sqrt [3]{a} e p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{b} d^2}+\frac {\left (\sqrt [3]{a} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^3}+\frac {\left (\sqrt [3]{a} e^2 p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^3}+\frac {\left (\sqrt [3]{a} e^2 p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^3} \\ & = \frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}+\frac {a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac {\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}-\frac {a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac {\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d^3}-\frac {3 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}+\frac {\left (3 a^{2/3} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac {\left (3 \sqrt [3]{a} e p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac {\left (e^3 p\right ) \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^3}-\frac {\left (e^3 p\right ) \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^3}-\frac {\left (e^3 p\right ) \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^3} \\ & = \frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\sqrt {3} a^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{a} e p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}+\frac {a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac {\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}-\frac {a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac {\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d^3}-\frac {3 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}-\frac {\left (e^2 p\right ) \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^3}-\frac {\left (e^2 p\right ) \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^3}-\frac {\left (e^2 p\right ) \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^3} \\ & = \frac {3 p}{4 d x^2}-\frac {3 e p}{d^2 x}-\frac {\sqrt {3} a^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{a} e p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}+\frac {a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac {\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}+\frac {e^2 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^3}-\frac {a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac {\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^3}\right )}{3 d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac {e^2 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^3}+\frac {e^2 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^3}-\frac {3 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.13 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.73 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {3 b e p \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {7}{3},-\frac {b}{a x^3}\right )}{4 a d^2 x^4}+\frac {3 b p \operatorname {Hypergeometric2F1}\left (1,\frac {5}{3},\frac {8}{3},-\frac {b}{a x^3}\right )}{10 a d x^5}-\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log \left (-\frac {b}{a x^3}\right )}{3 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac {3 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x^3}}{a}\right )}{3 d^3}-\frac {3 e^2 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.66 (sec) , antiderivative size = 489, normalized size of antiderivative = 0.66
method | result | size |
parts | \(-\frac {e^{2} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{d^{2} x}+\frac {3 p b \left (\frac {1}{2 d b \,x^{2}}-\frac {2 e}{d^{2} b x}+\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 d b \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 d b \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d b \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {2 e \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 d^{2} b \left (\frac {b}{a}\right )^{\frac {1}{3}}}-\frac {e \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{3 d^{2} b \left (\frac {b}{a}\right )^{\frac {1}{3}}}-\frac {2 e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d^{2} b \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {e^{2} \ln \left (x \right )^{2}}{d^{3} b}-\frac {2 e^{2} \left (\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 )\right )}{3 d^{3} b}-\frac {2 e^{2} \left (-\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 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^{3}}\right )}{2}\) | \(489\) |
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x^3}\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \]
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