Integrand size = 22, antiderivative size = 393 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {55 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a} \]
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Time = 0.24 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6326, 6330, 96, 94, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{11/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{56} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{48} a^5 c^4 x^6 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{48} a^4 c^4 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {55 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a}+\frac {55}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]
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Rule 94
Rule 96
Rule 214
Rule 6326
Rule 6330
Rubi steps \begin{align*} \text {integral}& = \left (a^8 c^4\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^4 x^8 \, dx \\ & = -\left (\left (a^8 c^4\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{11/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^{10}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {1}{9} \left (11 a^7 c^4\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^9} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{8} \left (11 a^6 c^4\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^8} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {1}{8} \left (11 a^5 c^4\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^7} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{48} \left (55 a^4 c^4\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^6} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {1}{16} \left (11 a^3 c^4\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{64} \left (11 a^2 c^4\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/2}}{x^4 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{192} \left (55 a c^4\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{128} \left (55 c^4\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {\left (55 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a} \\ & = \frac {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {\left (55 c^4\right ) \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 )}{128 a^2} \\ & = \frac {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {55 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.28 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-3712-4599 a x+10240 a^2 x^2-3066 a^3 x^3-8448 a^4 x^4+7224 a^5 x^5+1024 a^6 x^6-3024 a^7 x^7+896 a^8 x^8\right )+3465 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{8064 a} \]
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Time = 0.51 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\frac {\left (896 a^{8} x^{8}-3024 a^{7} x^{7}+1024 a^{6} x^{6}+7224 a^{5} x^{5}-8448 a^{4} x^{4}-3066 a^{3} x^{3}+10240 a^{2} x^{2}-4599 a x -3712\right ) \left (a x +1\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{8064 a}+\frac {55 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{128 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(160\) |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{4} \left (896 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}-3024 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}+1920 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+4200 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-6528 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+1134 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +8064 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-4352 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3465 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +3465 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{8064 a \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(288\) |
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Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.43 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (896 \, a^{9} c^{4} x^{9} - 2128 \, a^{8} c^{4} x^{8} - 2000 \, a^{7} c^{4} x^{7} + 8248 \, a^{6} c^{4} x^{6} - 1224 \, a^{5} c^{4} x^{5} - 11514 \, a^{4} c^{4} x^{4} + 7174 \, a^{3} c^{4} x^{3} + 5641 \, a^{2} c^{4} x^{2} - 8311 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8064 \, a} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=c^{4} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \frac {4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {6 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \frac {4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {4 a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \left (- \frac {a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{9} x^{9} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]
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Time = 0.20 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.06 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{8064} \, {\left (\frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (3465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{2}} - 30030 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 115038 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 334602 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 360448 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 255222 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 115038 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 30030 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3465 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {9 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {36 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {84 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {126 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {126 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {84 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {36 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {9 \, {\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac {{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \]
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Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.50 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {55 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{128 \, {\left | a \right |}} - \frac {1}{8064} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {3712 \, c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (4599 \, c^{4} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (5120 \, a c^{4} \mathrm {sgn}\left (a x + 1\right ) - {\left (1533 \, a^{2} c^{4} \mathrm {sgn}\left (a x + 1\right ) + 4 \, {\left (1056 \, a^{3} c^{4} \mathrm {sgn}\left (a x + 1\right ) - {\left (903 \, a^{4} c^{4} \mathrm {sgn}\left (a x + 1\right ) + 2 \, {\left (64 \, a^{5} c^{4} \mathrm {sgn}\left (a x + 1\right ) + 7 \, {\left (8 \, a^{7} c^{4} x \mathrm {sgn}\left (a x + 1\right ) - 27 \, a^{6} c^{4} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Time = 0.18 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.92 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {\frac {715\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96}-\frac {55\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64}-\frac {913\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32}+\frac {14179\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224}-\frac {5632\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{63}+\frac {18589\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{224}+\frac {913\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{32}-\frac {715\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{96}+\frac {55\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/2}}{64}}{a-\frac {9\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {36\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {84\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {126\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {126\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {84\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {36\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}+\frac {9\,a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}-\frac {a\,{\left (a\,x-1\right )}^9}{{\left (a\,x+1\right )}^9}}+\frac {55\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{64\,a} \]
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