Optimal. Leaf size=1304 \[ 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]{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 \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 (\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 {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]{-1} \sqrt [3]{d} 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]{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+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.17, 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 = {2500, 2526,
2498, 327, 206, 31, 648, 631, 210, 642, 2521, 2512, 266, 2463, 2437, 2338, 2441, 2440, 2438, 12}
\begin {gather*} -\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} \text {ArcTan}\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]{-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 (\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}}-\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 (-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 {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]{e}}-\frac {2 \sqrt [3]{d} \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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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]{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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 206
Rule 210
Rule 266
Rule 327
Rule 631
Rule 642
Rule 648
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2498
Rule 2500
Rule 2512
Rule 2521
Rule 2526
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 ) \text {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 ) \text {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 ) \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]{e}}-\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} 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]{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 ) \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]{e}}+\frac {\left (2 \sqrt [3]{d} 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]{e}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {\left (2 \sqrt [3]{-1} \sqrt [3]{d} 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]{e}}-\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} 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]{e}}+\frac {\left (2 (-1)^{2/3} \sqrt [3]{d} 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]{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 ) \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]{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 = 0.46, size = 1090, normalized size = 0.84 \begin {gather*} x \log ^2\left (c \left (d+e x^3\right )^p\right )-6 e p \left (-\frac {1}{2} p \left (\frac {6 x}{e}-\frac {\frac {2 \sqrt [3]{d} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\sqrt [3]{d} \left (\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e}}+\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}\right )}{e}\right )+\frac {x \log \left (c \left (d+e x^3\right )^p\right )}{e}-\frac {\sqrt [3]{d} \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^{4/3}}+\frac {\sqrt [3]{d} p \left (\frac {\log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {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 \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 \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 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}\right )}{6 e}+\frac {(-1)^{2/3} \sqrt [3]{d} p \left (\frac {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 {\log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {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 {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 \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 )}{\sqrt [3]{e}}\right )}{6 e}-\frac {\sqrt [3]{-1} \sqrt [3]{d} p \left (\frac {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 \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 {\log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {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]{e}}+\frac {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}}\right )}{6 e}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, 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(-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 \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{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 {\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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