Optimal. Leaf size=958 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.46813, antiderivative size = 958, normalized size of antiderivative = 1., number of steps used = 69, number of rules used = 21, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.75, Rules used = {6093, 2448, 321, 206, 2450, 2476, 2470, 12, 5984, 5918, 2402, 2315, 203, 2556, 5992, 5920, 2447, 4928, 4856, 4920, 4854} \[ x a^2+\frac{2 b \tan ^{-1}\left (\sqrt{c} x\right ) a}{\sqrt{c}}-\frac{2 b \tanh ^{-1}\left (\sqrt{c} x\right ) a}{\sqrt{c}}-b x \log \left (1-c x^2\right ) a+b x \log \left (c x^2+1\right ) a+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (c x^2+1\right )+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{i \sqrt{c} x+1}\right )}{\sqrt{c}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{\sqrt{c} x+1}\right )}{\sqrt{c}}+\frac{b^2 \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 \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 \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^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (c x^2+1\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (c x^2+1\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (c x^2+1\right )+\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}+\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1-i \sqrt{c} x}\right )}{\sqrt{c}}-\frac{i b^2 \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 \text{PolyLog}\left (2,1-\frac{2}{i \sqrt{c} x+1}\right )}{\sqrt{c}}+\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{\sqrt{c} x+1}\right )}{\sqrt{c}}-\frac{b^2 \text{PolyLog}\left (2,\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 \text{PolyLog}\left (2,1-\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 \text{PolyLog}\left (2,1-\frac{(1-i) \left (\sqrt{c} x+1\right )}{1-i \sqrt{c} x}\right )}{2 \sqrt{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6093
Rule 2448
Rule 321
Rule 206
Rule 2450
Rule 2476
Rule 2470
Rule 12
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 203
Rule 2556
Rule 5992
Rule 5920
Rule 2447
Rule 4928
Rule 4856
Rule 4920
Rule 4854
Rubi steps
\begin{align*} \int \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (a^2-a b \log \left (1-c x^2\right )+\frac{1}{4} b^2 \log ^2\left (1-c x^2\right )+a b \log \left (1+c x^2\right )-\frac{1}{2} b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 \log ^2\left (1+c x^2\right )\right ) \, dx\\ &=a^2 x-(a b) \int \log \left (1-c x^2\right ) \, dx+(a b) \int \log \left (1+c x^2\right ) \, dx+\frac{1}{4} b^2 \int \log ^2\left (1-c x^2\right ) \, dx+\frac{1}{4} b^2 \int \log ^2\left (1+c x^2\right ) \, dx-\frac{1}{2} b^2 \int \log \left (1-c x^2\right ) \log \left (1+c x^2\right ) \, dx\\ &=a^2 x-a b x \log \left (1-c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )+\frac{1}{2} b^2 \int \frac{2 c x^2 \log \left (1-c x^2\right )}{1+c x^2} \, dx+\frac{1}{2} b^2 \int -\frac{2 c x^2 \log \left (1+c x^2\right )}{1-c x^2} \, dx-(2 a b c) \int \frac{x^2}{1-c x^2} \, dx-(2 a b c) \int \frac{x^2}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac{x^2 \log \left (1-c x^2\right )}{1-c x^2} \, dx-\left (b^2 c\right ) \int \frac{x^2 \log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=a^2 x-a b x \log \left (1-c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )-(2 a b) \int \frac{1}{1-c x^2} \, dx+(2 a b) \int \frac{1}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac{x^2 \log \left (1-c x^2\right )}{1+c x^2} \, dx+\left (b^2 c\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\right ) \int \frac{x^2 \log \left (1+c x^2\right )}{1-c x^2} \, dx-\left (b^2 c\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\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-a b x \log \left (1-c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )-b^2 \int \log \left (1-c x^2\right ) \, dx+b^2 \int \frac{\log \left (1-c x^2\right )}{1-c x^2} \, dx-b^2 \int \log \left (1+c x^2\right ) \, dx+b^2 \int \frac{\log \left (1+c x^2\right )}{1+c x^2} \, dx+\left (b^2 c\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\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\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-a b x \log \left (1-c x^2\right )-b^2 x \log \left (1-c x^2\right )+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )-b^2 x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )+b^2 \int \log \left (1-c x^2\right ) \, dx-b^2 \int \frac{\log \left (1-c x^2\right )}{1+c x^2} \, dx+b^2 \int \log \left (1+c x^2\right ) \, dx-b^2 \int \frac{\log \left (1+c x^2\right )}{1-c x^2} \, dx-\left (2 b^2 c\right ) \int \frac{x^2}{1-c x^2} \, dx+\left (2 b^2 c\right ) \int \frac{x^2}{1+c x^2} \, dx-\left (2 b^2 c\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\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1-c x^2\right )} \, dx\\ &=a^2 x+4 b^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )-\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx-\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx-\left (2 b^2 \sqrt{c}\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx+\left (2 b^2 \sqrt{c}\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx+\left (2 b^2 c\right ) \int \frac{x^2}{1-c x^2} \, dx-\left (2 b^2 c\right ) \int \frac{x^2}{1+c x^2} \, dx-\left (2 b^2 c\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\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1+c x^2\right )} \, dx\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )+\left (2 b^2\right ) \int \frac{1}{1-c x^2} \, dx+\left (2 b^2\right ) \int \frac{1}{1+c x^2} \, dx+\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{i-\sqrt{c} x} \, dx+\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx-\left (2 b^2 \sqrt{c}\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx+\left (2 b^2 \sqrt{c}\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )-\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1-\sqrt{c} x}\right )}{1-c x^2} \, dx-\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1+i \sqrt{c} x}\right )}{1+c x^2} \, dx-\left (2 b^2 \sqrt{c}\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}\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\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}+\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )-b^2 \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx+b^2 \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{c} x} \, dx+\frac{\left (2 i b^2\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\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}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{-c} x} \, dx}{\sqrt{-c}}-\frac{\left (b^2 \sqrt{c}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{-c} x} \, dx}{\sqrt{-c}}\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \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 \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}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}+2 \left (b^2 \int \frac{\log \left (\frac{2}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx\right )-b^2 \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 (b^2 \int \frac{\log \left (\frac{2}{1+\sqrt{c} x}\right )}{1-c x^2} \, dx\right )-b^2 \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-b^2 \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-b^2 \int \frac{\log \left (\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \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 \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}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}-\frac{i b^2 \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 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}-\frac{b^2 \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 \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 \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\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{b^2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\sqrt{c} x}\right )}{\sqrt{c}}\\ &=a^2 x+\frac{2 a b \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}+\frac{i b^2 \tan ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}-\frac{2 a b \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right )^2}{\sqrt{c}}+\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}-\frac{2 b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \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 \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 \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}}-a b x \log \left (1-c x^2\right )-\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )}{\sqrt{c}}+\frac{1}{4} b^2 x \log ^2\left (1-c x^2\right )+a b x \log \left (1+c x^2\right )+\frac{b^2 \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{b^2 \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )}{\sqrt{c}}-\frac{1}{2} b^2 x \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac{1}{4} b^2 x \log ^2\left (1+c x^2\right )+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )}{\sqrt{c}}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1-i \sqrt{c} x}\right )}{\sqrt{c}}-\frac{i b^2 \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 \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )}{\sqrt{c}}+\frac{b^2 \text{Li}_2\left (1-\frac{2}{1+\sqrt{c} x}\right )}{\sqrt{c}}-\frac{b^2 \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 \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 \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*}
Mathematica [A] time = 2.56231, size = 566, normalized size = 0.59 \[ \frac{1}{2} x \left (\frac{b^2 \left (\text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{c x^2}\right )\right )-\text{PolyLog}\left (2,\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}-1\right )\right )-\text{PolyLog}\left (2,\left (-\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}-1\right )\right )-\text{PolyLog}\left (2,\frac{1}{2} \left (\sqrt{c x^2}+1\right )\right )+\text{PolyLog}\left (2,\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}+1\right )\right )+\text{PolyLog}\left (2,\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}+1\right )\right )-\frac{1}{2} i \text{PolyLog}\left (2,-e^{4 i \tan ^{-1}\left (\sqrt{c x^2}\right )}\right )+\frac{1}{2} \log ^2\left (1-\sqrt{c x^2}\right )-\frac{1}{2} \log ^2\left (\sqrt{c x^2}+1\right )-\log (2) \log \left (1-\sqrt{c x^2}\right )-\log \left (1-\sqrt{c x^2}\right ) \log \left (\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}-i\right )\right )+\log \left (\frac{1}{2} \left ((1+i)-(1-i) \sqrt{c x^2}\right )\right ) \log \left (\sqrt{c x^2}+1\right )+\log \left (\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}+i\right )\right ) \log \left (\sqrt{c x^2}+1\right )+\log (2) \log \left (\sqrt{c x^2}+1\right )-\log \left (1-\sqrt{c x^2}\right ) \log \left (\frac{1}{2} \left ((1-i) \sqrt{c x^2}+(1+i)\right )\right )-2 i \tan ^{-1}\left (\sqrt{c x^2}\right )^2+2 \sqrt{c x^2} \tanh ^{-1}\left (c x^2\right )^2+2 \tan ^{-1}\left (\sqrt{c x^2}\right ) \log \left (1+e^{4 i \tan ^{-1}\left (\sqrt{c x^2}\right )}\right )+2 \log \left (1-\sqrt{c x^2}\right ) \tanh ^{-1}\left (c x^2\right )-2 \log \left (\sqrt{c x^2}+1\right ) \tanh ^{-1}\left (c x^2\right )+4 \tan ^{-1}\left (\sqrt{c x^2}\right ) \tanh ^{-1}\left (c x^2\right )\right )}{\sqrt{c x^2}}+2 a^2+4 a b \tanh ^{-1}\left (c x^2\right )+\frac{4 a b \left (\tan ^{-1}\left (\sqrt{c x^2}\right )-\tanh ^{-1}\left (\sqrt{c x^2}\right )\right )}{\sqrt{c x^2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} \operatorname{artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{2}\right ) + a^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atanh}{\left (c x^{2} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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