Optimal. Leaf size=1304 \[ -\frac {\sqrt [3]{d} \log ^2\left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log ^2\left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log ^2\left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}+18 x p^2+\frac {6 \sqrt {3} \sqrt [3]{d} \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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]{e}}-\frac {2 \sqrt [3]{d} \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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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 (\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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\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]{e}}+\frac {3 \sqrt [3]{d} \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \text {Li}_2\left (\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \text {Li}_2\left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \text {Li}_2\left (\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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \text {Li}_2\left (-\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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \text {Li}_2\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]{e}}-6 x \log \left (c \left (e x^3+d\right )^p\right ) p+\frac {2 \sqrt [3]{d} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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]{e}}+x \log ^2\left (c \left (e x^3+d\right )^p\right ) \]
[Out]
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Rubi [A] time = 1.79, antiderivative size = 1310, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 20, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {2450, 2476, 2448, 321, 200, 31, 634, 617, 204, 628, 2471, 2462, 260, 2416, 2390, 2301, 2394, 2393, 2391, 12} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 200
Rule 204
Rule 260
Rule 321
Rule 617
Rule 628
Rule 634
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2448
Rule 2450
Rule 2462
Rule 2471
Rule 2476
Rubi steps
\begin {align*} \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx &=x \log ^2\left (c \left (d+e x^3\right )^p\right )-(6 e p) \int \frac {x^3 \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=x \log ^2\left (c \left (d+e x^3\right )^p\right )-(6 e p) \int \left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^3\right )^p\right )}{e \left (d+e x^3\right )}\right ) \, dx\\ &=x \log ^2\left (c \left (d+e x^3\right )^p\right )-(6 p) \int \log \left (c \left (d+e x^3\right )^p\right ) \, dx+(6 d p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=-6 p x \log \left (c \left (d+e x^3\right )^p\right )+x \log ^2\left (c \left (d+e x^3\right )^p\right )+(6 d p) \int \left (-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx+\left (18 e p^2\right ) \int \frac {x^3}{d+e x^3} \, dx\\ &=18 p^2 x-6 p x \log \left (c \left (d+e x^3\right )^p\right )+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (2 \sqrt [3]{d} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}-\sqrt [3]{e} x} \, dx-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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-\left (18 d p^2\right ) \int \frac {1}{d+e x^3} \, dx\\ &=18 p^2 x-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (6 \sqrt [3]{d} p^2\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (6 \sqrt [3]{d} p^2\right ) \int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx-\left (6 \sqrt [3]{d} e^{2/3} p^2\right ) \int \frac {x^2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{d+e x^3} \, dx+\left (6 \sqrt [3]{-1} \sqrt [3]{d} e^{2/3} 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-\left (6 (-1)^{2/3} \sqrt [3]{d} e^{2/3} 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\\ &=18 p^2 x-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (9 d^{2/3} p^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx+\frac {\left (3 \sqrt [3]{d} p^2\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{\sqrt [3]{e}}-\left (6 \sqrt [3]{d} e^{2/3} 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+\left (6 \sqrt [3]{-1} \sqrt [3]{d} e^{2/3} 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-\left (6 (-1)^{2/3} \sqrt [3]{d} e^{2/3} 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\\ &=18 p^2 x-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (2 \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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+\left (2 \sqrt [3]{-1} \sqrt [3]{d} 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+\left (2 \sqrt [3]{-1} \sqrt [3]{d} 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+\left (2 \sqrt [3]{-1} \sqrt [3]{d} 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-\left (2 (-1)^{2/3} \sqrt [3]{d} 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-\left (2 (-1)^{2/3} \sqrt [3]{d} 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-\left (2 (-1)^{2/3} \sqrt [3]{d} 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-\frac {\left (18 \sqrt [3]{d} p^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}}\\ &=18 p^2 x+\frac {6 \sqrt {3} \sqrt [3]{d} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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-\left (2 \sqrt [3]{d} 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-\frac {\left (2 \sqrt [3]{d} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}-\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}\\ &=18 p^2 x+\frac {6 \sqrt {3} \sqrt [3]{d} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {\sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )+\left (2 \sqrt [3]{-1} \sqrt [3]{d} 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+\frac {\left (2 \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}+\frac {\left (2 \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}-\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}\\ &=18 p^2 x+\frac {6 \sqrt {3} \sqrt [3]{d} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {\sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {(-1)^{2/3} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \operatorname {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]{e}}\\ &=18 p^2 x+\frac {6 \sqrt {3} \sqrt [3]{d} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {\sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {(-1)^{2/3} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 1090, normalized size = 0.84 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}, x\right ) \]
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 \[ \int \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.91, size = 0, normalized size = 0.00 \[ \int \ln \left (c \left (e \,x^{3}+d \right )^{p}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \log \left ({\left (e x^{3} + d\right )}^{p}\right )^{2} + \int \frac {e x^{3} \log \relax (c)^{2} + d \log \relax (c)^{2} - 2 \, {\left ({\left (3 \, e p - e \log \relax (c)\right )} x^{3} - d \log \relax (c)\right )} \log \left ({\left (e x^{3} + d\right )}^{p}\right )}{e x^{3} + d}\,{d x} \]
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 \[ \int {\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2 \,d x \]
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 \[ \int \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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