Optimal. Leaf size=1337 \[ \frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {4 b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {4 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{10 c^{5/2}}-\frac {b^2 \text {PolyLog}\left (2,-\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,-\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {i b^2 \text {PolyLog}\left (2,\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {PolyLog}\left (2,\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {b^2 \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{10 c^{5/2}}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )}{10 c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.04, antiderivative size = 1337, normalized size of antiderivative = 1.00, number of steps
used = 130, number of rules used = 31, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.938, Rules used =
{6045, 6042, 2507, 2526, 2505, 269, 331, 213, 212, 2520, 12, 266, 6820, 6135, 6079, 2497,
6874, 209, 30, 2637, 6139, 6031, 6057, 2449, 2352, 5048, 4940, 2438, 4966, 5044, 4988}
\begin {gather*} -\frac {i \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2 b^2}{5 c^{5/2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2 b^2}{5 c^{5/2}}-\frac {\log ^2\left (\frac {c}{x^2}+1\right ) b^2}{20 x^5}-\frac {4 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) b^2}{15 c^{5/2}}+\frac {4 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b^2}{15 c^{5/2}}+\frac {2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right ) b^2}{5 c^{5/2}}-\frac {\text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right ) b^2}{5 c^{5/2}}-\frac {\log \left (1-\frac {c}{x^2}\right ) b^2}{5 c^2 x}+\frac {\log \left (1-\frac {c}{x^2}\right ) b^2}{15 c x^3}-\frac {\log \left (1-\frac {c}{x^2}\right ) b^2}{25 x^5}+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{5 c^{5/2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{5 c^{5/2}}+\frac {\log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{10 x^5}-\frac {2 \log \left (\frac {c}{x^2}+1\right ) b^2}{15 c x^3}-\frac {2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right ) b^2}{5 c^{5/2}}+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right ) b^2}{5 c^{5/2}}+\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{x+\sqrt {c}}\right ) b^2}{5 c^{5/2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right ) b^2}{5 c^{5/2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right ) b^2}{5 c^{5/2}}+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) b^2}{5 c^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right ) b^2}{5 c^{5/2}}+\frac {i \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right ) b^2}{5 c^{5/2}}-\frac {i \text {Li}_2\left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}-1\right ) b^2}{5 c^{5/2}}-\frac {i \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right ) b^2}{10 c^{5/2}}-\frac {\text {Li}_2\left (-\frac {x}{\sqrt {c}}\right ) b^2}{5 c^{5/2}}-\frac {i \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right ) b^2}{5 c^{5/2}}+\frac {i \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right ) b^2}{5 c^{5/2}}+\frac {\text {Li}_2\left (\frac {x}{\sqrt {c}}\right ) b^2}{5 c^{5/2}}-\frac {\text {Li}_2\left (1-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right ) b^2}{5 c^{5/2}}+\frac {\text {Li}_2\left (\frac {2 \sqrt {c}}{x+\sqrt {c}}-1\right ) b^2}{5 c^{5/2}}+\frac {\text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right ) b^2}{10 c^{5/2}}+\frac {\text {Li}_2\left (1-\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right ) b^2}{10 c^{5/2}}-\frac {i \text {Li}_2\left (1-\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) b^2}{10 c^{5/2}}-\frac {8 b^2}{15 c^2 x}+\frac {2 a \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) b}{5 c^{5/2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{5 c^2 x}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{15 c x^3}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{25 x^5}-\frac {a \log \left (\frac {c}{x^2}+1\right ) b}{5 x^5}+\frac {2 a b}{5 c^2 x}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{25 x^5}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 209
Rule 212
Rule 213
Rule 266
Rule 269
Rule 331
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 2505
Rule 2507
Rule 2520
Rule 2526
Rule 2637
Rule 4940
Rule 4966
Rule 4988
Rule 5044
Rule 5048
Rule 6031
Rule 6042
Rule 6045
Rule 6057
Rule 6079
Rule 6135
Rule 6139
Rule 6820
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2}{x^6} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{4 x^6}-\frac {b \left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right )}{2 x^6}+\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{4 x^6}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{x^6} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right )}{x^6} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+\frac {c}{x^2}\right )}{x^6} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {1}{2} b \int \left (-\frac {2 a \log \left (1+\frac {c}{x^2}\right )}{x^6}+\frac {b \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{x^6}\right ) \, dx-\frac {1}{5} (b c) \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{\left (1-\frac {c}{x^2}\right ) x^8} \, dx-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\left (1+\frac {c}{x^2}\right ) x^8} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}+(a b) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^6} \, dx-\frac {1}{2} b^2 \int \frac {\log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{x^6} \, dx-\frac {1}{5} (b c) \int \left (-\frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{c x^6}-\frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{c^2 x^4}-\frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{c^3 x^2}-\frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{c^3 \left (c-x^2\right )}\right ) \, dx-\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1+\frac {c}{x^2}\right )}{c x^6}-\frac {\log \left (1+\frac {c}{x^2}\right )}{c^2 x^4}+\frac {\log \left (1+\frac {c}{x^2}\right )}{c^3 x^2}-\frac {\log \left (1+\frac {c}{x^2}\right )}{c^3 \left (c+x^2\right )}\right ) \, dx\\ &=-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}+\frac {1}{5} b \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{x^6} \, dx-\frac {1}{5} b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^6} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1-\frac {c}{x^2}\right )}{5 x^6 \left (c+x^2\right )} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+\frac {c}{x^2}\right )}{5 x^6 \left (c-x^2\right )} \, dx+\frac {b \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{x^2} \, dx}{5 c^2}+\frac {b \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{c-x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^2} \, dx}{5 c^2}+\frac {b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{c+x^2} \, dx}{5 c^2}+\frac {b \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{x^4} \, dx}{5 c}+\frac {b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^4} \, dx}{5 c}-\frac {1}{5} (2 a b c) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^8} \, dx\\ &=-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{25 x^5}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{5 c^2 x}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^6} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^6} \, dx-\frac {\left (2 b^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^4} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^4} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx}{5 c}-\frac {1}{5} (2 a b c) \int \frac {1}{x^6 \left (c+x^2\right )} \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^8} \, dx+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^8} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{x^6 \left (c+x^2\right )} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^6 \left (c-x^2\right )} \, dx\\ &=\frac {2 a b}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{25 x^5}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{5 c^2 x}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}+\frac {1}{5} (2 a b) \int \frac {1}{x^4 \left (c+x^2\right )} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{x^4 \left (-c+x^2\right )} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{x^4 \left (c+x^2\right )} \, dx+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx}{5 c^{3/2}}-\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (-c+x^2\right )} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{5 c}-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{x^6 \left (-c+x^2\right )} \, dx+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{x^6 \left (c+x^2\right )} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1-\frac {c}{x^2}\right )}{c x^6}-\frac {\log \left (1-\frac {c}{x^2}\right )}{c^2 x^4}+\frac {\log \left (1-\frac {c}{x^2}\right )}{c^3 x^2}-\frac {\log \left (1-\frac {c}{x^2}\right )}{c^3 \left (c+x^2\right )}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1+\frac {c}{x^2}\right )}{c x^6}+\frac {\log \left (1+\frac {c}{x^2}\right )}{c^2 x^4}+\frac {\log \left (1+\frac {c}{x^2}\right )}{c^3 x^2}+\frac {\log \left (1+\frac {c}{x^2}\right )}{c^3 \left (c-x^2\right )}\right ) \, dx\\ &=\frac {2 a b}{25 x^5}-\frac {4 b^2}{125 x^5}-\frac {2 a b}{15 c x^3}-\frac {4 b^2}{5 c^2 x}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{25 x^5}-\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \log \left (1+\frac {c}{x^2}\right )}{5 c^2 x}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {1}{25} \left (2 b^2\right ) \int \frac {1}{x^4 \left (-c+x^2\right )} \, dx-\frac {1}{25} \left (2 b^2\right ) \int \frac {1}{x^4 \left (c+x^2\right )} \, dx+\frac {1}{5} b^2 \int \frac {\log \left (1-\frac {c}{x^2}\right )}{x^6} \, dx+\frac {1}{5} b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^6} \, dx+\frac {b^2 \int \frac {\log \left (1-\frac {c}{x^2}\right )}{x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (1-\frac {c}{x^2}\right )}{c+x^2} \, dx}{5 c^2}+\frac {b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^2} \, dx}{5 c^2}+\frac {b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{c-x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{-c+x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{c+x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx}{5 c^{3/2}}-\frac {(2 a b) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (-c+x^2\right )} \, dx}{15 c}+\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{15 c}-\frac {b^2 \int \frac {\log \left (1-\frac {c}{x^2}\right )}{x^4} \, dx}{5 c}+\frac {b^2 \int \frac {\log \left (1+\frac {c}{x^2}\right )}{x^4} \, dx}{5 c}\\ &=\frac {2 a b}{25 x^5}-\frac {4 b^2}{125 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {16 b^2}{15 c^2 x}-\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^6} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^6} \, dx+\frac {\left (2 i b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (i+\frac {x}{\sqrt {c}}\right )} \, dx}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (1+\frac {x}{\sqrt {c}}\right )} \, dx}{5 c^{5/2}}+\frac {(2 a b) \int \frac {1}{c+x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{-c+x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{c+x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (-c+x^2\right )} \, dx}{25 c}+\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{25 c}+\frac {\left (2 b^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^4} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^4} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx}{5 c}+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^8} \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^8} \, dx\\ &=\frac {2 a b}{25 x^5}-\frac {4 b^2}{125 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {92 b^2}{75 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {8 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {8 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{x^4 \left (-c+x^2\right )} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{x^4 \left (c+x^2\right )} \, dx-\frac {\left (2 b^2\right ) \int \frac {\log \left (2-\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx}{5 c^3}+\frac {\left (2 b^2\right ) \int \frac {\log \left (2-\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx}{5 c^3}-\frac {\left (2 b^2\right ) \int \frac {1}{-c+x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{c+x^2} \, dx}{25 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (-c+x^2\right )} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{5 c}+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{x^6 \left (-c+x^2\right )} \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {1}{x^6 \left (c+x^2\right )} \, dx\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {32 b^2}{75 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {46 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{75 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {46 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {1}{25} \left (2 b^2\right ) \int \frac {1}{x^4 \left (-c+x^2\right )} \, dx+\frac {1}{25} \left (2 b^2\right ) \int \frac {1}{x^4 \left (c+x^2\right )} \, dx+\frac {\left (2 b^2\right ) \int \frac {1}{-c+x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {1}{c+x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx}{5 c^{3/2}}-\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (-c+x^2\right )} \, dx}{15 c}+\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{15 c}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {52 b^2}{75 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {16 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{75 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {16 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \frac {1}{-c+x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{c+x^2} \, dx}{15 c^2}+\frac {\left (2 b^2\right ) \int \left (-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c-x^2\right )}\right ) \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \left (\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c+x^2\right )}\right ) \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (-c+x^2\right )} \, dx}{25 c}-\frac {\left (2 b^2\right ) \int \frac {1}{x^2 \left (c+x^2\right )} \, dx}{25 c}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {26 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{75 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {26 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c-x^2} \, dx}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c+x^2} \, dx}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{-c+x^2} \, dx}{25 c^2}+\frac {\left (2 b^2\right ) \int \frac {1}{c+x^2} \, dx}{25 c^2}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {4 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {\left (i b^2\right ) \int \frac {\log \left (1-\frac {i x}{\sqrt {c}}\right )}{x} \, dx}{5 c^{5/2}}+\frac {\left (i b^2\right ) \int \frac {\log \left (1+\frac {i x}{\sqrt {c}}\right )}{x} \, dx}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx}{5 c^{5/2}}-\frac {\left (2 b^2\right ) \int \left (-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx}{5 c^{5/2}}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {4 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {b^2 \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx}{5 c^{5/2}}+\frac {b^2 \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx}{5 c^{5/2}}+\frac {b^2 \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}-x} \, dx}{5 c^{5/2}}-\frac {b^2 \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}+x} \, dx}{5 c^{5/2}}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {4 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+2 \frac {b^2 \int \frac {\log \left (\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx}{5 c^3}-\frac {b^2 \int \frac {\log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx}{5 c^3}-\frac {b^2 \int \frac {\log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx}{5 c^3}-2 \frac {b^2 \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx}{5 c^3}+\frac {b^2 \int \frac {\log \left (\frac {2 \left (\sqrt {-c}-x\right )}{\left (-1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx}{5 c^3}+\frac {b^2 \int \frac {\log \left (\frac {2 \left (\sqrt {-c}+x\right )}{\left (1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx}{5 c^3}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {4 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{10 c^{5/2}}-\frac {b^2 \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{10 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i 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-\frac {i x}{\sqrt {c}}}\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+\frac {x}{\sqrt {c}}}\right )}{5 c^{5/2}}\\ &=\frac {2 a b}{25 x^5}-\frac {2 a b}{15 c x^3}+\frac {2 a b}{5 c^2 x}-\frac {8 b^2}{15 c^2 x}+\frac {2 a b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {i b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {4 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{25 x^5}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{15 c x^3}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{5 c^2 x}-\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{5 c^{5/2}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{25 x^5}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{15 c x^3}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^2 x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{5 c^{5/2}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{20 x^5}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{5 x^5}-\frac {2 b^2 \log \left (1+\frac {c}{x^2}\right )}{15 c x^3}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{5 c^{5/2}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{10 x^5}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{20 x^5}-\frac {2 b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )}{10 c^{5/2}}-\frac {b^2 \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{10 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )}{10 c^{5/2}}\\ \end {align*}
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Mathematica [F]
time = 1.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}}{x^{6}}\, dx\]
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 (\frac {c}{x^{2}} \right )}\right )^{2}}{x^{6}}\, 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 (\frac {c}{x^2}\right )\right )}^2}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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