Optimal. Leaf size=1304 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.79105, antiderivative size = 1310, normalized size of antiderivative = 1., 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 2450
Rule 2476
Rule 2448
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 2471
Rule 2462
Rule 260
Rule 2416
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 12
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*}
Mathematica [A] time = 0.653003, size = 1090, normalized size = 0.84 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.059, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( e{x}^{3}+d \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]