Optimal. Leaf size=1085 \[ a^2 d x+\frac {2 a b d \text {ArcTan}\left (\sqrt {c} x\right )}{\sqrt {c}}+\frac {i b^2 d \text {ArcTan}\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}}+\frac {e \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2+\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 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 e \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \log \left (\frac {2}{1-c x^2}\right )}{c}-a b d x \log \left (1-c x^2\right )-\frac {b^2 d \text {ArcTan}\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 )+a b d x \log \left (1+c x^2\right )+\frac {b^2 d \text {ArcTan}\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 d x \log ^2\left (1+c x^2\right )+\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )}{\sqrt {c}}+\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}-\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt {c} x}\right )}{\sqrt {c}}-\frac {b^2 d \text {PolyLog}\left (2,1+\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 {PolyLog}\left (2,1-\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 {PolyLog}\left (2,1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{2 \sqrt {c}}-\frac {b^2 e \text {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.34, antiderivative size = 1085, normalized size of antiderivative = 1.00, number of steps
used = 77, number of rules used = 24, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6073, 6023,
2498, 327, 212, 2500, 2526, 2520, 12, 6131, 6055, 2449, 2352, 209, 2636, 6139, 6057, 2497, 5048,
4966, 5040, 4964, 6039, 6021} \begin {gather*} d x a^2+\frac {2 b d \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{\sqrt {c}}+\frac {b^2 d \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 2498
Rule 2500
Rule 2520
Rule 2526
Rule 2636
Rule 4964
Rule 4966
Rule 5040
Rule 5048
Rule 6021
Rule 6023
Rule 6039
Rule 6055
Rule 6057
Rule 6073
Rule 6131
Rule 6139
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 ) \text {Subst}\left (\int \log ^2(1-c x) \, dx,x,x^2\right )+\frac {1}{8} \left (b^2 e\right ) \text {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 ) \text {Subst}\left (\int \log ^2(x) \, dx,x,1-c x^2\right )}{8 c}+\frac {\left (b^2 e\right ) \text {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 ) \text {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{4 c}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{4 c}+\frac {1}{4} \left (b^2 c e\right ) \text {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 ) \text {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 ) \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 )-\frac {1}{4} \left (b^2 c e\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 )\\ &=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 ) \text {Subst}\left (\int \log (1-c x) \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 e\right ) \text {Subst}\left (\int \frac {\log (1-c x)}{1+c x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 e\right ) \text {Subst}\left (\int \log (1+c x) \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 e\right ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{4 c}+\frac {\left (b^2 e\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 2.13, size = 684, normalized size = 0.63 \begin {gather*} \frac {2 a^2 c d x^2+a^2 c e x^3+4 a b c d x^2 \tanh ^{-1}\left (c x^2\right )+4 a b d \sqrt {c x^2} \left (\text {ArcTan}\left (\sqrt {c x^2}\right )-\tanh ^{-1}\left (\sqrt {c x^2}\right )\right )+b^2 e x \tanh ^{-1}\left (c x^2\right ) \left (\left (-1+c x^2\right ) \tanh ^{-1}\left (c x^2\right )-2 \log \left (1+e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )\right )+a b e x \left (2 c x^2 \tanh ^{-1}\left (c x^2\right )+\log \left (1-c^2 x^4\right )\right )+b^2 e x \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )-b^2 d \sqrt {c x^2} \left (2 i \text {ArcTan}\left (\sqrt {c x^2}\right )^2-4 \text {ArcTan}\left (\sqrt {c x^2}\right ) \tanh ^{-1}\left (c x^2\right )-2 \sqrt {c x^2} \tanh ^{-1}\left (c x^2\right )^2-2 \text {ArcTan}\left (\sqrt {c x^2}\right ) \log \left (1+e^{4 i \text {ArcTan}\left (\sqrt {c x^2}\right )}\right )-2 \tanh ^{-1}\left (c x^2\right ) \log \left (1-\sqrt {c x^2}\right )+\log (2) \log \left (1-\sqrt {c x^2}\right )-\frac {1}{2} \log ^2\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 (-i+\sqrt {c x^2}\right )\right )+2 \tanh ^{-1}\left (c x^2\right ) \log \left (1+\sqrt {c x^2}\right )-\log (2) \log \left (1+\sqrt {c x^2}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {c x^2}\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {c x^2}\right )\right )+\frac {1}{2} i \text {PolyLog}\left (2,-e^{4 i \text {ArcTan}\left (\sqrt {c x^2}\right )}\right )-\text {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {c x^2}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )+\text {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {c x^2}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )\right )}{2 c x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, 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 \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 \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2} \left (d + e x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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