Optimal. Leaf size=1173 \[ \frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \text {ArcTan}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \text {ArcTan}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \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 )}{10 c^{5/2}}-\frac {b^2 \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 )}{10 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.64, antiderivative size = 1173, normalized size of antiderivative = 1.00, number of steps
used = 102, number of rules used = 26, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.625, Rules used = {6041, 2507,
2526, 2498, 327, 212, 2505, 308, 2520, 12, 6131, 6055, 2449, 2352, 6874, 209, 30, 2637, 213, 6139,
6057, 2497, 5048, 4966, 5040, 4964} \begin {gather*} \frac {1}{20} \left (2 a-b \log \left (1-c x^2\right )\right )^2 x^5+\frac {1}{20} b^2 \log ^2\left (c x^2+1\right ) x^5-\frac {2}{25} a b x^5+\frac {1}{25} b^2 \log \left (1-c x^2\right ) x^5+\frac {1}{25} b \left (2 a-b \log \left (1-c x^2\right )\right ) x^5+\frac {1}{5} a b \log \left (c x^2+1\right ) x^5-\frac {1}{10} b^2 \log \left (1-c x^2\right ) \log \left (c x^2+1\right ) x^5-\frac {b^2 \log \left (1-c x^2\right ) x^3}{15 c}+\frac {b \left (2 a-b \log \left (1-c x^2\right )\right ) x^3}{15 c}+\frac {2 b^2 \log \left (c x^2+1\right ) x^3}{15 c}+\frac {2 a b x^3}{15 c}+\frac {8 b^2 x}{15 c^2}+\frac {i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \text {ArcTan}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {2 a b \text {ArcTan}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right )}{5 c^{5/2}}-\frac {b^2 \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 )}{10 c^{5/2}}-\frac {b^2 \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 )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 209
Rule 212
Rule 213
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 2498
Rule 2505
Rule 2507
Rule 2520
Rule 2526
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 x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {1}{2} b x^4 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 x^4 \log ^2\left (1+c x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \, dx-\frac {1}{2} b \int x^4 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right ) \, dx+\frac {1}{4} b^2 \int x^4 \log ^2\left (1+c x^2\right ) \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{2} b \int \left (-2 a x^4 \log \left (1+c x^2\right )+b x^4 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )\right ) \, dx-\frac {1}{5} (b c) \int \frac {x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{1-c x^2} \, dx-\frac {1}{5} \left (b^2 c\right ) \int \frac {x^6 \log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+(a b) \int x^4 \log \left (1+c x^2\right ) \, dx-\frac {1}{2} b^2 \int x^4 \log \left (1-c x^2\right ) \log \left (1+c x^2\right ) \, dx-\frac {1}{5} (b c) \int \left (-\frac {2 a-b \log \left (1-c x^2\right )}{c^3}-\frac {x^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{c^2}-\frac {x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{c}+\frac {2 a-b \log \left (1-c x^2\right )}{c^3 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{c^3}-\frac {x^2 \log \left (1+c x^2\right )}{c^2}+\frac {x^4 \log \left (1+c x^2\right )}{c}-\frac {\log \left (1+c x^2\right )}{c^3 \left (1+c x^2\right )}\right ) \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {1}{5} b \int x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx-\frac {1}{5} b^2 \int x^4 \log \left (1+c x^2\right ) \, dx+\frac {1}{2} b^2 \int \frac {2 c x^6 \log \left (1-c x^2\right )}{5+5 c x^2} \, dx+\frac {1}{2} b^2 \int \frac {2 c x^6 \log \left (1+c x^2\right )}{-5+5 c x^2} \, dx+\frac {b \int \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx}{5 c^2}-\frac {b \int \frac {2 a-b \log \left (1-c x^2\right )}{1-c x^2} \, dx}{5 c^2}-\frac {b^2 \int \log \left (1+c x^2\right ) \, dx}{5 c^2}+\frac {b^2 \int \frac {\log \left (1+c x^2\right )}{1+c x^2} \, dx}{5 c^2}+\frac {b \int x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx}{5 c}+\frac {b^2 \int x^2 \log \left (1+c x^2\right ) \, dx}{5 c}-\frac {1}{5} (2 a b c) \int \frac {x^6}{1+c x^2} \, dx\\ &=\frac {2 a b x}{5 c^2}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac {b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1-c x^2} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1+c x^2} \, dx-\frac {b^2 \int \log \left (1-c x^2\right ) \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {x^2}{1+c x^2} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx}{5 c}-\frac {1}{5} (2 a b c) \int \left (\frac {1}{c^3}-\frac {x^2}{c^2}+\frac {x^4}{c}-\frac {1}{c^3 \left (1+c x^2\right )}\right ) \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1-c x^2} \, dx+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac {x^6 \log \left (1-c x^2\right )}{5+5 c x^2} \, dx+\left (b^2 c\right ) \int \frac {x^6 \log \left (1+c x^2\right )}{-5+5 c x^2} \, dx\\ &=\frac {2 b^2 x}{5 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5-\frac {b^2 x \log \left (1-c x^2\right )}{5 c^2}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac {b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}-\frac {x^2}{c}+\frac {1}{c^2 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}+\frac {x^2}{c}+\frac {1}{c^2 \left (1+c x^2\right )}\right ) \, dx+\frac {(2 a b) \int \frac {1}{1+c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx}{5 c^{3/2}}-\frac {\left (2 b^2\right ) \int \frac {x^2}{1-c x^2} \, dx}{5 c}-\frac {1}{25} \left (2 b^2 c\right ) \int \left (-\frac {1}{c^3}-\frac {x^2}{c^2}-\frac {x^4}{c}+\frac {1}{c^3 \left (1-c x^2\right )}\right ) \, dx+\frac {1}{25} \left (2 b^2 c\right ) \int \left (\frac {1}{c^3}-\frac {x^2}{c^2}+\frac {x^4}{c}-\frac {1}{c^3 \left (1+c x^2\right )}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac {\log \left (1-c x^2\right )}{5 c^3}-\frac {x^2 \log \left (1-c x^2\right )}{5 c^2}+\frac {x^4 \log \left (1-c x^2\right )}{5 c}-\frac {\log \left (1-c x^2\right )}{c^3 \left (5+5 c x^2\right )}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{5 c^3}+\frac {x^2 \log \left (1+c x^2\right )}{5 c^2}+\frac {x^4 \log \left (1+c x^2\right )}{5 c}+\frac {\log \left (1+c x^2\right )}{c^3 \left (-5+5 c x^2\right )}\right ) \, dx\\ &=\frac {92 b^2 x}{75 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {4 b^2 x^5}{125}+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {b^2 x \log \left (1-c x^2\right )}{5 c^2}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac {b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {1}{5} b^2 \int x^4 \log \left (1-c x^2\right ) \, dx+\frac {1}{5} b^2 \int x^4 \log \left (1+c x^2\right ) \, dx-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{15 c^2}+\frac {b^2 \int \log \left (1-c x^2\right ) \, dx}{5 c^2}+\frac {b^2 \int \log \left (1+c x^2\right ) \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{i-\sqrt {c} x} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (1-c x^2\right )}{5+5 c x^2} \, dx}{c^2}+\frac {b^2 \int \frac {\log \left (1+c x^2\right )}{-5+5 c x^2} \, dx}{c^2}-\frac {b^2 \int x^2 \log \left (1-c x^2\right ) \, dx}{5 c}+\frac {b^2 \int x^2 \log \left (1+c x^2\right ) \, dx}{5 c}\\ &=\frac {92 b^2 x}{75 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {4 b^2 x^5}{125}+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {46 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {46 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1-c x^2} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1+c x^2} \, dx-\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-\sqrt {c} x}\right )}{1-c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1+i \sqrt {c} x}\right )}{1+c x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {x^2}{1-c x^2} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {x^2}{1+c x^2} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{5 \sqrt {c} \left (1-c x^2\right )} \, dx}{c}-\frac {\left (2 b^2\right ) \int -\frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{5 \sqrt {c} \left (1+c x^2\right )} \, dx}{c}+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1-c x^2} \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1+c x^2} \, dx\\ &=\frac {32 b^2 x}{75 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {4 b^2 x^5}{125}+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {46 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {46 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}-\frac {x^2}{c}+\frac {1}{c^2 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}+\frac {x^2}{c}+\frac {1}{c^2 \left (1+c x^2\right )}\right ) \, dx+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx}{5 c^{3/2}}+\frac {1}{25} \left (2 b^2 c\right ) \int \left (-\frac {1}{c^3}-\frac {x^2}{c^2}-\frac {x^4}{c}+\frac {1}{c^3 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \left (\frac {1}{c^3}-\frac {x^2}{c^2}+\frac {x^4}{c}-\frac {1}{c^3 \left (1+c x^2\right )}\right ) \, dx\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {16 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {16 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{25 c^2}+\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{15 c^2}-\frac {\left (2 b^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}{5 c^{3/2}}+\frac {\left (2 b^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}{5 c^{3/2}}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx}{5 c^2}+\frac {b^2 \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {c} x} \, dx}{5 c^2}+\frac {b^2 \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {-c} x} \, dx}{5 \sqrt {-c} c^{3/2}}-\frac {b^2 \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {-c} x} \, dx}{5 \sqrt {-c} c^{3/2}}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+2 \frac {b^2 \int \frac {\log \left (\frac {2}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx}{5 c^2}-\frac {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}{5 c^2}+2 \frac {b^2 \int \frac {\log \left (\frac {2}{1+\sqrt {c} x}\right )}{1-c x^2} \, dx}{5 c^2}-\frac {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}{5 c^2}-\frac {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}{5 c^2}-\frac {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}{5 c^2}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\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 )}{10 c^{5/2}}-\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 )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+2 \frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+2 \frac {b^2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt {c} x}\right )}{5 c^{5/2}}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}+\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 )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}-\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 )}{10 c^{5/2}}-\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 )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}\\ \end {align*}
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Mathematica [F]
time = 8.45, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{4} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}\, dx\]
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 x^{4} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{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 x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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