Optimal. Leaf size=1170 \[ -\frac {e^{2/3} \log ^2\left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{2 d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} \log ^2\left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{2 d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} \log ^2\left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{2 d^{2/3}}-\frac {e^{2/3} \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}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} \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}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} \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}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} \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}{d^{2/3}}-\frac {e^{2/3} \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}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} \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}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} \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}{d^{2/3}}-\frac {e^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} \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}{d^{2/3}}-\frac {e^{2/3} \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}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} \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}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} \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}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} \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}{d^{2/3}}+\frac {e^{2/3} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} \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}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} \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}{d^{2/3}}-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{2 x^2} \]
[Out]
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Rubi [A] time = 1.34, antiderivative size = 1176, 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 = {2457, 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 260
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2457
Rule 2462
Rule 2471
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}+(3 e p) \int \frac {\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 )}{2 x^2}+(3 e 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\\ &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {(e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}-\sqrt [3]{e} x} \, dx}{d^{2/3}}-\frac {(e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{d^{2/3}}-\frac {(e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x} \, dx}{d^{2/3}}\\ &=\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {\left (3 e^{5/3} p^2\right ) \int \frac {x^2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{d+e x^3} \, dx}{d^{2/3}}+\frac {\left (3 \sqrt [3]{-1} e^{5/3} 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}{d^{2/3}}-\frac {\left (3 (-1)^{2/3} e^{5/3} 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}{d^{2/3}}\\ &=\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {\left (3 e^{5/3} 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}{d^{2/3}}+\frac {\left (3 \sqrt [3]{-1} e^{5/3} 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}{d^{2/3}}-\frac {\left (3 (-1)^{2/3} e^{5/3} 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}{d^{2/3}}\\ &=\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {\left (e p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}+\frac {\left (\sqrt [3]{-1} e 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}{d^{2/3}}+\frac {\left (\sqrt [3]{-1} e 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}{d^{2/3}}+\frac {\left (\sqrt [3]{-1} e 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}{d^{2/3}}-\frac {\left ((-1)^{2/3} e 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}{d^{2/3}}-\frac {\left ((-1)^{2/3} e 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}{d^{2/3}}-\frac {\left ((-1)^{2/3} e 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}{d^{2/3}}\\ &=-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {\left (e^{2/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {\left (\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\left ((-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}-\frac {\left (e 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}{d^{2/3}}\\ &=-\frac {e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{2 d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}+\frac {\left (e^{2/3} 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 )}{d^{2/3}}+\frac {\left (e^{2/3} 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 )}{d^{2/3}}+\frac {\left (\sqrt [3]{-1} e^{2/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{d^{2/3}}-\frac {\left (\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\left ((-1)^{2/3} e^{2/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d^{2/3}}+\frac {\left ((-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {\left ((-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {\left (\sqrt [3]{-1} e 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}{d^{2/3}}\\ &=-\frac {e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{2 d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {e^{2/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\left (\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}\\ &=-\frac {e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{2 d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} p^2 \log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}+\frac {e^{2/3} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{d^{2/3}}+\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {e^{2/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{d^{2/3}}-\frac {e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {(-1)^{2/3} e^{2/3} 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 )}{d^{2/3}}-\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}+\frac {\sqrt [3]{-1} e^{2/3} 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 )}{d^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 745, normalized size = 0.64 \[ \frac {1}{2} \left (-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2}+\frac {e^{2/3} 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 (-1)^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 \sqrt [3]{-1} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-(-1)^{2/3} p \left (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]{-1} \sqrt [3]{e} x-\sqrt [3]{d}}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\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]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )+2 \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left ((-1)^{2/3}-1\right ) \sqrt [3]{d}}\right )\right )\right )+\sqrt [3]{-1} p \left (2 \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 )+2 \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+\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)^{2/3}-1\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 )\right )-p \left (2 \text {Li}_2\left (\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {2 i \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{3 i+\sqrt {3}}\right )+\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 {-\frac {2 i \sqrt [3]{e} x}{\sqrt [3]{d}}+\sqrt {3}+i}{\sqrt {3}+3 i}\right )\right )\right )\right )\right )}{d^{2/3}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.98, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{x^{3}}\, dx \]
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 \[ -\frac {\log \left ({\left (e x^{3} + d\right )}^{p}\right )^{2}}{2 \, x^{2}} + \int \frac {e x^{3} \log \relax (c)^{2} + d \log \relax (c)^{2} + {\left ({\left (3 \, e p + 2 \, e \log \relax (c)\right )} x^{3} + 2 \, d \log \relax (c)\right )} \log \left ({\left (e x^{3} + d\right )}^{p}\right )}{e x^{6} + d x^{3}}\,{d x} \]
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 \[ \int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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