Optimal. Leaf size=1085 \[ d x a^2+\frac {2 b d \tan ^{-1}\left (\sqrt {c} x\right ) a}{\sqrt {c}}-\frac {2 b d \tanh ^{-1}\left (\sqrt {c} x\right ) a}{\sqrt {c}}-b d x \log \left (1-c x^2\right ) a+b d x \log \left (c x^2+1\right ) a+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2+\frac {e \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{2 c}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (c x^2+1\right )+\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b e \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \log \left (\frac {2}{1-c x^2}\right )}{c}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{\sqrt {c}}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (c x^2+1\right )+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 d \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {b^2 d \text {Li}_2\left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}+1\right )}{2 \sqrt {c}}-\frac {b^2 d \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{2 \sqrt {c}}-\frac {i b^2 d \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}-\frac {b^2 e \text {Li}_2\left (1-\frac {2}{1-c x^2}\right )}{2 c} \]
[Out]
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Rubi [A] time = 2.44, antiderivative size = 1216, normalized size of antiderivative = 1.12, number of steps used = 104, number of rules used = 39, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.167, Rules used = {6742, 6091, 298, 203, 206, 6097, 260, 6093, 2450, 2476, 2448, 321, 2470, 12, 5984, 5918, 2402, 2315, 2556, 5992, 5920, 2447, 4928, 4856, 4920, 4854, 6099, 2454, 2389, 2296, 2295, 30, 2557, 2475, 43, 2416, 2394, 2393, 2391} \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 30
Rule 43
Rule 203
Rule 206
Rule 260
Rule 298
Rule 321
Rule 2295
Rule 2296
Rule 2315
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2402
Rule 2416
Rule 2447
Rule 2448
Rule 2450
Rule 2454
Rule 2470
Rule 2475
Rule 2476
Rule 2556
Rule 2557
Rule 4854
Rule 4856
Rule 4920
Rule 4928
Rule 5918
Rule 5920
Rule 5984
Rule 5992
Rule 6091
Rule 6093
Rule 6097
Rule 6099
Rule 6742
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (a^2 (d+e x)+2 a b (d+e x) \tanh ^{-1}\left (c x^2\right )+b^2 (d+e x) \tanh ^{-1}\left (c x^2\right )^2\right ) \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+(2 a b) \int (d+e x) \tanh ^{-1}\left (c x^2\right ) \, dx+b^2 \int (d+e x) \tanh ^{-1}\left (c x^2\right )^2 \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+(2 a b) \int \left (d \tanh ^{-1}\left (c x^2\right )+e x \tanh ^{-1}\left (c x^2\right )\right ) \, dx+b^2 \int \left (d \tanh ^{-1}\left (c x^2\right )^2+e x \tanh ^{-1}\left (c x^2\right )^2\right ) \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+(2 a b d) \int \tanh ^{-1}\left (c x^2\right ) \, dx+\left (b^2 d\right ) \int \tanh ^{-1}\left (c x^2\right )^2 \, dx+(2 a b e) \int x \tanh ^{-1}\left (c x^2\right ) \, dx+\left (b^2 e\right ) \int x \tanh ^{-1}\left (c x^2\right )^2 \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\left (b^2 d\right ) \int \left (\frac {1}{4} \log ^2\left (1-c x^2\right )-\frac {1}{2} \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} \log ^2\left (1+c x^2\right )\right ) \, dx-(4 a b c d) \int \frac {x^2}{1-c^2 x^4} \, dx+\left (b^2 e\right ) \int \left (\frac {1}{4} x \log ^2\left (1-c x^2\right )-\frac {1}{2} x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} x \log ^2\left (1+c x^2\right )\right ) \, dx-(2 a b c e) \int \frac {x^3}{1-c^2 x^4} \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-(2 a b d) \int \frac {1}{1-c x^2} \, dx+(2 a b d) \int \frac {1}{1+c x^2} \, dx+\frac {1}{4} \left (b^2 d\right ) \int \log ^2\left (1-c x^2\right ) \, dx+\frac {1}{4} \left (b^2 d\right ) \int \log ^2\left (1+c x^2\right ) \, dx-\frac {1}{2} \left (b^2 d\right ) \int \log \left (1-c x^2\right ) \log \left (1+c x^2\right ) \, dx+\frac {1}{4} \left (b^2 e\right ) \int x \log ^2\left (1-c x^2\right ) \, dx+\frac {1}{4} \left (b^2 e\right ) \int x \log ^2\left (1+c x^2\right ) \, dx-\frac {1}{2} \left (b^2 e\right ) \int x \log \left (1-c x^2\right ) \log \left (1+c x^2\right ) \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}+\frac {1}{2} \left (b^2 d\right ) \int \frac {2 c x^2 \log \left (1-c x^2\right )}{1+c x^2} \, dx+\frac {1}{2} \left (b^2 d\right ) \int -\frac {2 c x^2 \log \left (1+c x^2\right )}{1-c x^2} \, dx+\left (b^2 c d\right ) \int \frac {x^2 \log \left (1-c x^2\right )}{1-c x^2} \, dx-\left (b^2 c d\right ) \int \frac {x^2 \log \left (1+c x^2\right )}{1+c x^2} \, dx+\frac {1}{8} \left (b^2 e\right ) \operatorname {Subst}\left (\int \log ^2(1-c x) \, dx,x,x^2\right )+\frac {1}{8} \left (b^2 e\right ) \operatorname {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^2\right )+\frac {1}{2} \left (b^2 e\right ) \int \frac {c x^3 \log \left (1-c x^2\right )}{1+c x^2} \, dx-\frac {1}{2} \left (b^2 e\right ) \int \frac {c x^3 \log \left (1+c x^2\right )}{1-c x^2} \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}+\left (b^2 c d\right ) \int \frac {x^2 \log \left (1-c x^2\right )}{1+c x^2} \, dx+\left (b^2 c d\right ) \int \left (-\frac {\log \left (1-c x^2\right )}{c}+\frac {\log \left (1-c x^2\right )}{c \left (1-c x^2\right )}\right ) \, dx-\left (b^2 c d\right ) \int \frac {x^2 \log \left (1+c x^2\right )}{1-c x^2} \, dx-\left (b^2 c d\right ) \int \left (\frac {\log \left (1+c x^2\right )}{c}-\frac {\log \left (1+c x^2\right )}{c \left (1+c x^2\right )}\right ) \, dx-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1-c x^2\right )}{8 c}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{8 c}+\frac {1}{2} \left (b^2 c e\right ) \int \frac {x^3 \log \left (1-c x^2\right )}{1+c x^2} \, dx-\frac {1}{2} \left (b^2 c e\right ) \int \frac {x^3 \log \left (1+c x^2\right )}{1-c x^2} \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\left (b^2 d\right ) \int \log \left (1-c x^2\right ) \, dx+\left (b^2 d\right ) \int \frac {\log \left (1-c x^2\right )}{1-c x^2} \, dx-\left (b^2 d\right ) \int \log \left (1+c x^2\right ) \, dx+\left (b^2 d\right ) \int \frac {\log \left (1+c x^2\right )}{1+c x^2} \, dx+\left (b^2 c d\right ) \int \left (\frac {\log \left (1-c x^2\right )}{c}-\frac {\log \left (1-c x^2\right )}{c \left (1+c x^2\right )}\right ) \, dx-\left (b^2 c d\right ) \int \left (-\frac {\log \left (1+c x^2\right )}{c}+\frac {\log \left (1+c x^2\right )}{c \left (1-c x^2\right )}\right ) \, dx+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{4 c}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{4 c}+\frac {1}{4} \left (b^2 c e\right ) \operatorname {Subst}\left (\int \frac {x \log (1-c x)}{1+c x} \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 c e\right ) \operatorname {Subst}\left (\int \frac {x \log (1+c x)}{1-c x} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 e x^2+\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )-b^2 d x \log \left (1-c x^2\right )+\frac {b^2 e \left (1-c x^2\right ) \log \left (1-c x^2\right )}{4 c}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-b^2 d x \log \left (1+c x^2\right )-\frac {b^2 e \left (1+c x^2\right ) \log \left (1+c x^2\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}+\left (b^2 d\right ) \int \log \left (1-c x^2\right ) \, dx-\left (b^2 d\right ) \int \frac {\log \left (1-c x^2\right )}{1+c x^2} \, dx+\left (b^2 d\right ) \int \log \left (1+c x^2\right ) \, dx-\left (b^2 d\right ) \int \frac {\log \left (1+c x^2\right )}{1-c x^2} \, dx-\left (2 b^2 c d\right ) \int \frac {x^2}{1-c x^2} \, dx+\left (2 b^2 c d\right ) \int \frac {x^2}{1+c x^2} \, dx-\left (2 b^2 c d\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx+\left (2 b^2 c d\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx+\frac {1}{4} \left (b^2 c e\right ) \operatorname {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 )-\frac {1}{4} \left (b^2 c e\right ) \operatorname {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 )\\ &=4 b^2 d x+\frac {1}{2} b^2 e x^2+\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {b^2 e \left (1-c x^2\right ) \log \left (1-c x^2\right )}{4 c}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \left (1+c x^2\right ) \log \left (1+c x^2\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\left (2 b^2 d\right ) \int \frac {1}{1-c x^2} \, dx-\left (2 b^2 d\right ) \int \frac {1}{1+c x^2} \, dx-\left (2 b^2 \sqrt {c} d\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx+\left (2 b^2 \sqrt {c} d\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx+\left (2 b^2 c d\right ) \int \frac {x^2}{1-c x^2} \, dx-\left (2 b^2 c d\right ) \int \frac {x^2}{1+c x^2} \, dx-\left (2 b^2 c d\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx+\left (2 b^2 c d\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx+\frac {1}{4} \left (b^2 e\right ) \operatorname {Subst}\left (\int \log (1-c x) \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\log (1-c x)}{1+c x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 e\right ) \operatorname {Subst}\left (\int \log (1+c x) \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^2 e x^2+\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {b^2 e \left (1-c x^2\right ) \log \left (1-c x^2\right )}{4 c}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \log \left (1-c x^2\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}-\frac {b^2 e \left (1+c x^2\right ) \log \left (1+c x^2\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}+\frac {b^2 e \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{4 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}+\left (2 b^2 d\right ) \int \frac {1}{1-c x^2} \, dx+\left (2 b^2 d\right ) \int \frac {1}{1+c x^2} \, dx+\left (2 b^2 d\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{i-\sqrt {c} x} \, dx+\left (2 b^2 d\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx-\left (2 b^2 \sqrt {c} d\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx+\left (2 b^2 \sqrt {c} d\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx-\frac {1}{4} \left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^2\right )-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{4 c}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{4 c}\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \log \left (1-c x^2\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}+\frac {b^2 e \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{4 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1-\sqrt {c} x}\right )}{1-c x^2} \, dx-\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i \sqrt {c} x}\right )}{1+c x^2} \, dx-\left (2 b^2 \sqrt {c} d\right ) \int \left (\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1-\sqrt {c} x\right )}-\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1+\sqrt {c} x\right )}\right ) \, dx+\left (2 b^2 \sqrt {c} d\right ) \int \left (-\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1-\sqrt {-c} x\right )}+\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1+\sqrt {-c} x\right )}\right ) \, dx+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{4 c}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{4 c}\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \log \left (1-c x^2\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}+\frac {b^2 e \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{4 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1-c x^2\right )\right )}{4 c}+\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}-\left (b^2 d\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx+\left (b^2 d\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {c} x} \, dx+\frac {\left (2 i b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\left (2 b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {\left (b^2 \sqrt {c} d\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {-c} x} \, dx}{\sqrt {-c}}-\frac {\left (b^2 \sqrt {c} d\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {-c} x} \, dx}{\sqrt {-c}}\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \log \left (1-c x^2\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}+\frac {b^2 e \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{4 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1-c x^2\right )\right )}{4 c}+\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}+2 \left (\left (b^2 d\right ) \int \frac {\log \left (\frac {2}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\right )-\left (b^2 d\right ) \int \frac {\log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx+2 \left (\left (b^2 d\right ) \int \frac {\log \left (\frac {2}{1+\sqrt {c} x}\right )}{1-c x^2} \, dx\right )-\left (b^2 d\right ) \int \frac {\log \left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (-\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx-\left (b^2 d\right ) \int \frac {\log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx-\left (b^2 d\right ) \int \frac {\log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \log \left (1-c x^2\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}+\frac {b^2 e \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{4 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1-c x^2\right )\right )}{4 c}+\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 d \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{2 \sqrt {c}}-\frac {b^2 d \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{2 \sqrt {c}}-\frac {i b^2 d \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+2 \frac {\left (i b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+2 \frac {\left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt {c} x}\right )}{\sqrt {c}}\\ &=\frac {a^2 (d+e x)^2}{2 e}+\frac {2 a b d \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \tan ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}-\frac {2 a b d \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right )^2}{\sqrt {c}}+2 a b d x \tanh ^{-1}\left (c x^2\right )+a b e x^2 \tanh ^{-1}\left (c x^2\right )+\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{\sqrt {c}}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{\sqrt {c}}+\frac {1}{4} b^2 d x \log ^2\left (1-c x^2\right )-\frac {b^2 e \left (1-c x^2\right ) \log ^2\left (1-c x^2\right )}{8 c}-\frac {b^2 e \log \left (1-c x^2\right ) \log \left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}-\frac {b^2 d \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{\sqrt {c}}+\frac {b^2 e \log \left (\frac {1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{4 c}-\frac {1}{2} b^2 d x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )-\frac {1}{4} b^2 e x^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 d x \log ^2\left (1+c x^2\right )+\frac {b^2 e \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac {a b e \log \left (1-c^2 x^4\right )}{2 c}-\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1-c x^2\right )\right )}{4 c}+\frac {b^2 e \text {Li}_2\left (\frac {1}{2} \left (1+c x^2\right )\right )}{4 c}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 d \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{2 \sqrt {c}}-\frac {b^2 d \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{2 \sqrt {c}}-\frac {i b^2 d \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 3.00, size = 684, normalized size = 0.63 \[ \frac {2 a^2 c d x^2+a^2 c e x^3+a b e x \left (\log \left (1-c^2 x^4\right )+2 c x^2 \tanh ^{-1}\left (c x^2\right )\right )+4 a b c d x^2 \tanh ^{-1}\left (c x^2\right )+4 a b d \sqrt {c x^2} \left (\tan ^{-1}\left (\sqrt {c x^2}\right )-\tanh ^{-1}\left (\sqrt {c x^2}\right )\right )-b^2 d \sqrt {c x^2} \left (-\text {Li}_2\left (\frac {1}{2} \left (1-\sqrt {c x^2}\right )\right )+\text {Li}_2\left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {c x^2}-1\right )\right )+\text {Li}_2\left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {c x^2}-1\right )\right )+\text {Li}_2\left (\frac {1}{2} \left (\sqrt {c x^2}+1\right )\right )-\text {Li}_2\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {c x^2}+1\right )\right )-\text {Li}_2\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {c x^2}+1\right )\right )+\frac {1}{2} i \text {Li}_2\left (-e^{4 i \tan ^{-1}\left (\sqrt {c x^2}\right )}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {c x^2}\right )+\frac {1}{2} \log ^2\left (\sqrt {c x^2}+1\right )+\log (2) \log \left (1-\sqrt {c x^2}\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {c x^2}-i\right )\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {c x^2}\right )\right ) \log \left (\sqrt {c x^2}+1\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\sqrt {c x^2}+i\right )\right ) \log \left (\sqrt {c x^2}+1\right )-\log (2) \log \left (\sqrt {c x^2}+1\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\frac {1}{2} \left ((1-i) \sqrt {c x^2}+(1+i)\right )\right )+2 i \tan ^{-1}\left (\sqrt {c x^2}\right )^2-2 \sqrt {c x^2} \tanh ^{-1}\left (c x^2\right )^2-2 \tan ^{-1}\left (\sqrt {c x^2}\right ) \log \left (1+e^{4 i \tan ^{-1}\left (\sqrt {c x^2}\right )}\right )-2 \log \left (1-\sqrt {c x^2}\right ) \tanh ^{-1}\left (c x^2\right )+2 \log \left (\sqrt {c x^2}+1\right ) \tanh ^{-1}\left (c x^2\right )-4 \tan ^{-1}\left (\sqrt {c x^2}\right ) \tanh ^{-1}\left (c x^2\right )\right )+b^2 e x \text {Li}_2\left (-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )+b^2 e x \tanh ^{-1}\left (c x^2\right ) \left (\left (c x^2-1\right ) \tanh ^{-1}\left (c x^2\right )-2 \log \left (e^{-2 \tanh ^{-1}\left (c x^2\right )}+1\right )\right )}{2 c x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} e x + a^{2} d + {\left (b^{2} e x + b^{2} d\right )} \operatorname {artanh}\left (c x^{2}\right )^{2} + 2 \, {\left (a b e x + a b d\right )} \operatorname {artanh}\left (c x^{2}\right ), x\right ) \]
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 \[ \int {\left (e x + d\right )} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a^{2} e x^{2} + {\left (c {\left (\frac {2 \, \arctan \left (\sqrt {c} x\right )}{c^{\frac {3}{2}}} + \frac {\log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} + 2 \, x \operatorname {artanh}\left (c x^{2}\right )\right )} a b d + a^{2} d x + \frac {{\left (2 \, c x^{2} \operatorname {artanh}\left (c x^{2}\right ) + \log \left (-c^{2} x^{4} + 1\right )\right )} a b e}{2 \, c} + \frac {1}{8} \, {\left (b^{2} e x^{2} + 2 \, b^{2} d x\right )} \log \left (-c x^{2} + 1\right )^{2} - \int -\frac {{\left (b^{2} c e x^{3} + b^{2} c d x^{2} - b^{2} e x - b^{2} d\right )} \log \left (c x^{2} + 1\right )^{2} - 2 \, {\left (b^{2} c e x^{3} + 2 \, b^{2} c d x^{2} + {\left (b^{2} c e x^{3} + b^{2} c d x^{2} - b^{2} e x - b^{2} d\right )} \log \left (c x^{2} + 1\right )\right )} \log \left (-c x^{2} + 1\right )}{4 \, {\left (c x^{2} - 1\right )}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2} \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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