Optimal. Leaf size=1137 \[ \frac {\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\sqrt [3]{-1} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {(-1)^{2/3} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {2 \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text {Li}_2\left (-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.90, antiderivative size = 1143, normalized size of antiderivative = 1.01, number of steps
used = 39, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {2507, 2526,
2512, 266, 2463, 2437, 2338, 2441, 2440, 2438, 12} \begin {gather*} \frac {\sqrt [3]{e} \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{\sqrt [3]{d}}-\frac {\sqrt [3]{-1} \sqrt [3]{e} \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{\sqrt [3]{d}}+\frac {(-1)^{2/3} \sqrt [3]{e} \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (-\frac {\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} \text {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} \text {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} \text {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} \text {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} \text {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{e} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 266
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2507
Rule 2512
Rule 2526
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+(6 e p) \int \frac {x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+(6 e p) \int \left (-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {(-1)^{2/3} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx\\ &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}-\frac {\left (2 e^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 \sqrt [3]{-1} e^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 (-1)^{2/3} e^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {\left (6 e^{4/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx}{\sqrt [3]{d}}-\frac {\left (6 \sqrt [3]{-1} e^{4/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d+e x^3} \, dx}{\sqrt [3]{d}}+\frac {\left (6 (-1)^{2/3} e^{4/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{d+e x^3} \, dx}{\sqrt [3]{d}}\\ &=-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {\left (6 e^{4/3} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt [3]{d}}-\frac {\left (6 \sqrt [3]{-1} e^{4/3} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt [3]{d}}+\frac {\left (6 (-1)^{2/3} e^{4/3} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{\sqrt [3]{d}}\\ &=-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {\left (2 e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {\left (2 \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\left (2 \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1} \log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {\left (2 \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {(-1)^{2/3} \log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\left (2 e^{2/3} p^2\right ) \int \frac {\log \left (\frac {\sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 e^{2/3} p^2\right ) \int \frac {\log \left (\frac {\sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac {\log \left (\frac {(-1)^{2/3} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}+\frac {\left (2 \sqrt [3]{-1} e^{2/3} p^2\right ) \int \frac {\log \left (\frac {(-1)^{2/3} \sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac {\log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}-\frac {\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac {\log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=\frac {\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}-\frac {\left (2 \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\left (2 \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\left (2 (-1)^{2/3} \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {\left (2 (-1)^{2/3} e^{2/3} p^2\right ) \int \frac {\log \left (\frac {\sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{d}}\\ &=\frac {\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\sqrt [3]{-1} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {(-1)^{2/3} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {2 \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text {Li}_2\left (-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{e} p^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}\\ &=\frac {\sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {\sqrt [3]{-1} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}+\frac {(-1)^{2/3} \sqrt [3]{e} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{e} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+\frac {2 \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text {Li}_2\left (-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}-\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}+\frac {2 (-1)^{2/3} \sqrt [3]{e} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{d}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 742, normalized size = 0.65 \begin {gather*} -\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}-\frac {\sqrt [3]{e} p \left (2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 \sqrt [3]{-1} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )+2 (-1)^{2/3} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )+\sqrt [3]{-1} p \left (\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \left (2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )+2 \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )-(-1)^{2/3} p \left (\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \left (2 \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )-p \left (\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \left (\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \left (\log \left (\frac {\sqrt [3]{-1} \sqrt [3]{d}-\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{e} x}{\sqrt [3]{d}}}{3 i+\sqrt {3}}\right )\right )\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {2 i \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 i+\sqrt {3}}\right )\right )\right )}{\sqrt [3]{d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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