Optimal. Leaf size=942 \[ 2 a b \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-2 b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-b^2 \sqrt {c} \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 )-b^2 \sqrt {c} \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 )+b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )+i b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+i b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1+i \sqrt {c} x}\right )-b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt {c} x}\right )+\frac {1}{2} b^2 \sqrt {c} \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 )+\frac {1}{2} b^2 \sqrt {c} \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 )-\frac {1}{2} i b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.95, antiderivative size = 942, normalized size of antiderivative = 1.00, number of steps
used = 47, number of rules used = 21, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.313, Rules used = {6041, 2507,
212, 2520, 12, 6131, 6055, 2449, 2352, 2505, 6874, 209, 30, 2637, 6139, 6057, 2497, 5048, 4966,
5040, 4964} \begin {gather*} i \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right )^2 b^2+\sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2 b^2-\frac {\log ^2\left (c x^2+1\right ) b^2}{4 x}-2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right ) b^2-2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right ) b^2+\sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) b^2+2 \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right ) b^2+2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right ) b^2-\sqrt {c} \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 ) b^2-\sqrt {c} \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 ) b^2+\sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) b^2-\sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right ) b^2+\sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) b^2+\sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) b^2+\frac {\log \left (1-c x^2\right ) \log \left (c x^2+1\right ) b^2}{2 x}-\sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right ) b^2+i \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right ) b^2-\frac {1}{2} i \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) b^2+i \sqrt {c} \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right ) b^2-\sqrt {c} \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right ) b^2+\frac {1}{2} \sqrt {c} \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 ) b^2+\frac {1}{2} \sqrt {c} \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 ) b^2-\frac {1}{2} i \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) b^2+2 a \sqrt {c} \text {ArcTan}\left (\sqrt {c} x\right ) b+\sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) b-\frac {a \log \left (c x^2+1\right ) b}{x}-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 209
Rule 212
Rule 2352
Rule 2449
Rule 2497
Rule 2505
Rule 2507
Rule 2520
Rule 2637
Rule 4964
Rule 4966
Rule 5040
Rule 5048
Rule 6041
Rule 6055
Rule 6057
Rule 6131
Rule 6139
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^2} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x^2}-\frac {b \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{2 x^2}+\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{x^2} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{x^2} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+c x^2\right )}{x^2} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-\frac {1}{2} b \int \left (-\frac {2 a \log \left (1+c x^2\right )}{x^2}+\frac {b \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^2}\right ) \, dx+(b c) \int \frac {2 a-b \log \left (1-c x^2\right )}{1-c x^2} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}+(a b) \int \frac {\log \left (1+c x^2\right )}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^2} \, dx-\left (2 b^2 c^2\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^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx\\ &=b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}+\frac {1}{2} b^2 \int -\frac {2 c \log \left (1-c x^2\right )}{1+c x^2} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+c x^2\right )}{1-c x^2} \, dx+(2 a b c) \int \frac {1}{1+c x^2} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-\left (b^2 c\right ) \int \frac {\log \left (1-c x^2\right )}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{1-c x^2} \, dx+\left (2 b^2 c\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{i-\sqrt {c} x} \, dx-\left (2 b^2 c\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}+\left (2 b^2 c\right ) \int \frac {\log \left (\frac {2}{1-\sqrt {c} x}\right )}{1-c x^2} \, dx-\left (2 b^2 c\right ) \int \frac {\log \left (\frac {2}{1+i \sqrt {c} x}\right )}{1+c x^2} \, dx-\left (2 b^2 c^2\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^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}+\left (2 i b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i \sqrt {c} x}\right )-\left (2 b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {c} x}\right )-\left (2 b^2 c^{3/2}\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-\left (2 b^2 c^{3/2}\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 c^{3/2}\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\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-\left (b^2 c\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx+\left (b^2 c\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {c} x} \, dx-\frac {\left (b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {-c} x} \, dx}{\sqrt {-c}}+\frac {\left (b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {-c} x} \, dx}{\sqrt {-c}}\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-b^2 \sqrt {c} \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 )-b^2 \sqrt {c} \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 )+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+2 \left (\left (b^2 c\right ) \int \frac {\log \left (\frac {2}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\right )-\left (b^2 c\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 c\right ) \int \frac {\log \left (\frac {2}{1+\sqrt {c} x}\right )}{1-c x^2} \, dx\right )+\left (b^2 c\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 c\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 c\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 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-b^2 \sqrt {c} \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 )-b^2 \sqrt {c} \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 )+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+\frac {1}{2} b^2 \sqrt {c} \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 )+\frac {1}{2} b^2 \sqrt {c} \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 )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+2 \left (\left (i b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i \sqrt {c} x}\right )\right )-2 \left (\left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt {c} x}\right )\right )\\ &=2 a b \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )+i b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right )^2+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right )^2-2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+2 b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-b^2 \sqrt {c} \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 )-b^2 \sqrt {c} \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 )+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )+b \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac {a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt {c} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+\sqrt {c} x}\right )+\frac {1}{2} b^2 \sqrt {c} \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 )+\frac {1}{2} b^2 \sqrt {c} \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 )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )\\ \end {align*}
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Mathematica [A]
time = 2.32, size = 566, normalized size = 0.60 \begin {gather*} \frac {-2 a^2-4 a b \tanh ^{-1}\left (c x^2\right )+4 a b \sqrt {c x^2} \left (\text {ArcTan}\left (\sqrt {c x^2}\right )+\tanh ^{-1}\left (\sqrt {c x^2}\right )\right )+b^2 \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 )-\frac {2 \tanh ^{-1}\left (c x^2\right )^2}{\sqrt {c x^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 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{x^{2}}\, dx\]
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 \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{2}}\, 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 \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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