Integrand size = 20, antiderivative size = 328 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \]
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Time = 0.20 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6329, 105, 157, 12, 94, 214} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}} \]
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Rule 12
Rule 94
Rule 105
Rule 157
Rule 214
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^4} \\ & = \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{a}-\frac {7 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {a \text {Subst}\left (\int \frac {\frac {7}{a^2}+\frac {48 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^2 \text {Subst}\left (\int \frac {-\frac {35}{a^3}-\frac {275 x}{a^4}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{35 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {a^3 \text {Subst}\left (\int \frac {\frac {105}{a^4}+\frac {1240 x}{a^5}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{105 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^4 \text {Subst}\left (\int \frac {-\frac {105}{a^5}-\frac {4035 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{105 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^5 \text {Subst}\left (\int \frac {-\frac {525}{a^6}-\frac {7860 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{525 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^6 \text {Subst}\left (\int \frac {-\frac {1575}{a^7}-\frac {7335 x}{a^8}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{1575 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^7 \text {Subst}\left (\int -\frac {1575}{a^8 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{1575 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c^4} \\ & = -\frac {8}{7 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {11}{7 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {62}{21 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {269}{21 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.35 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (384-279 a x-1065 a^2 x^2+715 a^3 x^3+965 a^4 x^4-559 a^5 x^5-281 a^6 x^6+105 a^7 x^7\right )}{105 (-1+a x)^4 (1+a x)^3}+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \]
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Time = 0.26 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{56 a^{13} \left (x -\frac {1}{a}\right )^{4}}-\frac {17 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{112 a^{12} \left (x -\frac {1}{a}\right )^{3}}-\frac {211 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{336 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {1657 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{672 a^{10} \left (x -\frac {1}{a}\right )}-\frac {7 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{60 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {379 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{480 a^{10} \left (x +\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{80 a^{12} \left (x +\frac {1}{a}\right )^{3}}\right ) a^{8} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(366\) |
default | \(\text {Expression too large to display}\) | \(898\) |
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Time = 0.26 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (105 \, a^{7} x^{7} - 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} + 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} - 1065 \, a^{2} x^{2} - 279 \, a x + 384\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {a^{8} \int \frac {x^{8}}{a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {1}{6720} \, a {\left (\frac {5 \, {\left (\frac {39 \, {\left (a x - 1\right )}}{a x + 1} + \frac {287 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2611 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {5628 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 3\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {7 \, {\left (3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 50 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 705 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {47\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {41\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {373\,{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {268\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {13\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{7}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}-\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^4} \]
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