Optimal. Leaf size=141 \[ \frac {3 b \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 c^2}+\frac {3 b x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 c}-\frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{4 c^2}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \log \left (\frac {2}{1-c x^2}\right )}{2 c^2}-\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )}{4 c^2} \]
[Out]
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Rubi [A]
time = 0.22, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6039, 6037,
6127, 6021, 6131, 6055, 2449, 2352, 6095} \begin {gather*} -\frac {3 b^2 \log \left (\frac {2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{2 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{4 c^2}+\frac {3 b \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 c^2}+\frac {3 b x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x^2}\right )}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6039
Rule 6055
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^3 \left (2 a-b \log \left (1-c x^2\right )\right )^3+\frac {3}{8} b x^3 \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )-\frac {3}{8} b^2 x^3 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac {1}{8} b^3 x^3 \log ^3\left (1+c x^2\right )\right ) \, dx\\ &=\frac {1}{8} \int x^3 \left (2 a-b \log \left (1-c x^2\right )\right )^3 \, dx+\frac {1}{8} (3 b) \int x^3 \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right ) \, dx-\frac {1}{8} \left (3 b^2\right ) \int x^3 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right ) \, dx+\frac {1}{8} b^3 \int x^3 \log ^3\left (1+c x^2\right ) \, dx\\ &=\frac {1}{16} \text {Subst}\left (\int x (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )+\frac {1}{16} (3 b) \text {Subst}\left (\int x (-2 a+b \log (1-c x))^2 \log (1+c x) \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int x (-2 a+b \log (1-c x)) \log ^2(1+c x) \, dx,x,x^2\right )+\frac {1}{16} b^3 \text {Subst}\left (\int x \log ^3(1+c x) \, dx,x,x^2\right )\\ &=\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac {1}{16} \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x))^3}{c}-\frac {(1-c x) (2 a-b \log (1-c x))^3}{c}\right ) \, dx,x,x^2\right )+\frac {1}{16} b^3 \text {Subst}\left (\int \left (-\frac {\log ^3(1+c x)}{c}+\frac {(1+c x) \log ^3(1+c x)}{c}\right ) \, dx,x,x^2\right )-\frac {1}{32} (3 b c) \text {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^2\right )+\frac {1}{16} \left (3 b^2 c\right ) \text {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,x^2\right )+\frac {1}{16} \left (3 b^2 c\right ) \text {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^2\right )-\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {x^2 \log ^2(1+c x)}{1-c x} \, dx,x,x^2\right )\\ &=\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac {\text {Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )}{16 c}-\frac {\text {Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )}{16 c}-\frac {b^3 \text {Subst}\left (\int \log ^3(1+c x) \, dx,x,x^2\right )}{16 c}+\frac {b^3 \text {Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,x^2\right )}{16 c}-\frac {1}{32} (3 b c) \text {Subst}\left (\int \left (-\frac {(-2 a+b \log (1-c x))^2}{c^2}+\frac {x (-2 a+b \log (1-c x))^2}{c}+\frac {(-2 a+b \log (1-c x))^2}{c^2 (1+c x)}\right ) \, dx,x,x^2\right )+\frac {1}{16} \left (3 b^2 c\right ) \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2}+\frac {x (2 a-b \log (1-c x)) \log (1+c x)}{c}+\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^2\right )+\frac {1}{16} \left (3 b^2 c\right ) \text {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2}-\frac {x (2 a-b \log (1-c x)) \log (1+c x)}{c}-\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2 (1+c x)}\right ) \, dx,x,x^2\right )-\frac {1}{32} \left (3 b^3 c\right ) \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c^2}-\frac {x \log ^2(1+c x)}{c}-\frac {\log ^2(1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {1}{32} (3 b) \text {Subst}\left (\int x (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )+\frac {1}{32} \left (3 b^3\right ) \text {Subst}\left (\int x \log ^2(1+c x) \, dx,x,x^2\right )-\frac {\text {Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-c x^2\right )}{16 c^2}+\frac {\text {Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-c x^2\right )}{16 c^2}-\frac {b^3 \text {Subst}\left (\int \log ^3(x) \, dx,x,1+c x^2\right )}{16 c^2}+\frac {b^3 \text {Subst}\left (\int x \log ^3(x) \, dx,x,1+c x^2\right )}{16 c^2}+\frac {(3 b) \text {Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )}{32 c}-\frac {(3 b) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^2\right )}{32 c}+2 \frac {\left (3 b^2\right ) \text {Subst}\left (\int (2 a-b \log (1-c x)) \log (1+c x) \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {(2 a-b \log (1-c x)) \log (1+c x)}{-1+c x} \, dx,x,x^2\right )}{16 c}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {(2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^2\right )}{32 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{-1+c x} \, dx,x,x^2\right )}{32 c}\\ &=-\frac {\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac {\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{32 c^2}+\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3 b^3 \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac {b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}-\frac {1}{32} (3 b) \text {Subst}\left (\int \left (\frac {(-2 a+b \log (1-c x))^2}{c}-\frac {(1-c x) (-2 a+b \log (1-c x))^2}{c}\right ) \, dx,x,x^2\right )+\frac {1}{32} \left (3 b^3\right ) \text {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c}+\frac {(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,x^2\right )+2 \left (\frac {3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \frac {x (2 a-b \log (1-c x))}{1+c x} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^3\right ) \text {Subst}\left (\int \frac {x \log (1+c x)}{1-c x} \, dx,x,x^2\right )\right )+\frac {(3 b) \text {Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}-\frac {(3 b) \text {Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}-\frac {(3 b) \text {Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^2\right )}{16 c^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )}{16 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^2\right )}{16 c}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right ) \log (1+c x)}{1+c x} \, dx,x,x^2\right )}{16 c}\\ &=-\frac {9 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 c^2}+\frac {3 b \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2}{64 c^2}-\frac {\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac {\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{32 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {9 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 c^2}-\frac {3 b^3 \left (1+c x^2\right )^2 \log ^2\left (1+c x^2\right )}{64 c^2}+\frac {3 b^3 \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}-\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac {b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+2 \left (\frac {3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac {1}{16} \left (3 b^2\right ) \text {Subst}\left (\int \left (\frac {2 a-b \log (1-c x)}{c}-\frac {2 a-b \log (1-c x)}{c (1+c x)}\right ) \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^3\right ) \text {Subst}\left (\int \left (-\frac {\log (1+c x)}{c}-\frac {\log (1+c x)}{c (-1+c x)}\right ) \, dx,x,x^2\right )\right )-\frac {(3 b) \text {Subst}\left (\int \frac {(2 a-b \log (x))^2}{2-x} \, dx,x,1-c x^2\right )}{32 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-c x^2\right )}{32 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{16 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2-x}{2}\right ) (-2 a+b \log (x))}{x} \, dx,x,1-c x^2\right )}{16 c^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^2\right )}{8 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int x \log (x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{2-x} \, dx,x,1+c x^2\right )}{32 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2-x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{8 c^2}-\frac {(3 b) \text {Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )}{32 c}+\frac {(3 b) \text {Subst}\left (\int (1-c x) (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )}{32 c}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^2\right )}{32 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,x^2\right )}{32 c}\\ &=\frac {9 a b^2 x^2}{8 c}+\frac {9 b^3 x^2}{16 c}+\frac {3 b^3 \left (1-c x^2\right )^2}{128 c^2}-\frac {3 b^3 \left (1+c x^2\right )^2}{128 c^2}+\frac {3 b^2 \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{64 c^2}-\frac {9 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 c^2}+\frac {3 b \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2}{64 c^2}-\frac {\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac {\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac {9 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^3 \left (1+c x^2\right )^2 \log \left (1+c x^2\right )}{64 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {9 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 c^2}-\frac {3 b^3 \left (1+c x^2\right )^2 \log ^2\left (1+c x^2\right )}{64 c^2}-\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac {b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^2\right )\right )}{16 c^2}+\frac {3 b^3 \log \left (1+c x^2\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac {(3 b) \text {Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}-\frac {(3 b) \text {Subst}\left (\int x (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{8 c^2}+2 \left (\frac {3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int (2 a-b \log (1-c x)) \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {2 a-b \log (1-c x)}{1+c x} \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (1+c x) \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac {9 a b^2 x^2}{8 c}+\frac {9 b^3 x^2}{8 c}+\frac {3 b^3 \left (1-c x^2\right )^2}{128 c^2}-\frac {3 b^3 \left (1+c x^2\right )^2}{128 c^2}+\frac {9 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^2}+\frac {3 b^2 \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{64 c^2}-\frac {3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac {\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac {\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac {9 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^3 \left (1+c x^2\right )^2 \log \left (1+c x^2\right )}{64 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c^2}-\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac {b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+\frac {3 b^3 \text {Li}_3\left (\frac {1}{2} \left (1-c x^2\right )\right )}{16 c^2}-\frac {3 b^3 \text {Li}_3\left (\frac {1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int x (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{32 c^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int x \log (x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^2}+2 \left (-\frac {3 a b^2 x^2}{8 c}+\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac {3 b^3 \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (1-c x) \, dx,x,x^2\right )}{16 c}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac {3 a b^2 x^2}{4 c}+\frac {15 b^3 x^2}{16 c}+\frac {9 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^2}-\frac {3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac {\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac {\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac {3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{8 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c^2}-\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac {b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+2 \left (-\frac {3 a b^2 x^2}{8 c}-\frac {3 b^3 x^2}{16 c}+\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac {3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^3 \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^2}\right )-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^2}\\ &=\frac {3 a b^2 x^2}{4 c}+\frac {3 b^3 x^2}{4 c}+\frac {3 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{8 c^2}-\frac {3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac {\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac {\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac {3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{8 c^2}-\frac {3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac {3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c^2}-\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac {3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac {b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac {b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+2 \left (-\frac {3 a b^2 x^2}{8 c}-\frac {3 b^3 x^2}{8 c}-\frac {3 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^2}+\frac {3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac {3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^3 \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac {3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac {3 b^3 \text {Li}_2\left (\frac {1}{2} \left (1-c x^2\right )\right )}{16 c^2}+\frac {3 b^3 \text {Li}_2\left (\frac {1}{2} \left (1+c x^2\right )\right )}{16 c^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 185, normalized size = 1.31 \begin {gather*} \frac {6 b^2 \left (-1+c x^2\right ) \left (a+b+a c x^2\right ) \tanh ^{-1}\left (c x^2\right )^2+2 b^3 \left (-1+c^2 x^4\right ) \tanh ^{-1}\left (c x^2\right )^3+6 b \tanh ^{-1}\left (c x^2\right ) \left (a c x^2 \left (2 b+a c x^2\right )-2 b^2 \log \left (1+e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )\right )+a \left (6 a b c x^2+2 a^2 c^2 x^4+3 a b \log \left (1-c x^2\right )-3 a b \log \left (1+c x^2\right )+6 b^2 \log \left (1-c^2 x^4\right )\right )+6 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )}{8 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.32, size = 798, normalized size = 5.66
method | result | size |
risch | \(\text {Expression too large to display}\) | \(798\) |
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 \begin {gather*} \text {Failed to integrate} \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 x^{3} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{3}\, 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 = -0.01 \begin {gather*} \int x^3\,{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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