Integrand size = 20, antiderivative size = 268 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {15 c^3 \csc ^{-1}(a x)}{8 a}+\frac {c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
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Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6329, 99, 159, 163, 41, 222, 94, 214} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}{5 a}+c^3 x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{20 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{60 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{24 a}-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{8 a}+\frac {15 c^3 \csc ^{-1}(a x)}{8 a} \]
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Rule 41
Rule 94
Rule 99
Rule 159
Rule 163
Rule 214
Rule 222
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-c^3 \text {Subst}\left (\int \frac {\left (\frac {1}{a}-\frac {6 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{5} \left (a c^3\right ) \text {Subst}\left (\int \frac {\left (\frac {5}{a^2}-\frac {23 x}{a^3}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{20} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {\left (\frac {20}{a^3}-\frac {43 x}{a^4}\right ) \left (1+\frac {x}{a}\right )^{5/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {1}{60} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {60}{a^4}+\frac {155 x}{a^5}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{120} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (\frac {120}{a^5}-\frac {345 x}{a^6}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {1}{120} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {-\frac {120}{a^6}+\frac {225 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {\left (15 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}-\frac {c^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = -\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {c^3 \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}+\frac {\left (15 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = -\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {15 c^3 \csc ^{-1}(a x)}{8 a}+\frac {c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.39 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (-24-30 a x+88 a^2 x^2+135 a^3 x^3-184 a^4 x^4+120 a^5 x^5\right )}{x^4}+225 a^4 \arcsin \left (\frac {1}{a x}\right )+120 a^4 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{120 a^5} \]
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Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {\left (a x -1\right ) \left (184 a^{4} x^{4}-135 a^{3} x^{3}-88 a^{2} x^{2}+30 a x +24\right ) c^{3}}{120 x^{5} a^{6} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {a^{6} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {15 a^{5} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}+a^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{6} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(172\) |
default | \(\frac {\left (a x -1\right ) c^{3} \left (-120 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+120 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+225 a^{5} x^{5} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}+225 a^{5} x^{5} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-105 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-64 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{120 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{5} \sqrt {a^{2}}}\) | \(272\) |
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Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {450 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (120 \, a^{6} c^{3} x^{6} - 64 \, a^{5} c^{3} x^{5} - 49 \, a^{4} c^{3} x^{4} + 223 \, a^{3} c^{3} x^{3} + 58 \, a^{2} c^{3} x^{2} - 54 \, a c^{3} x - 24 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{120 \, a^{6} x^{5}} \]
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\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {a^{6}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {3 a^{2}}{x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {3 a^{4}}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.13 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {1}{60} \, {\left (\frac {225 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {345 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 1345 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 1654 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 86 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 305 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 105 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {5 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac {4 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \]
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Time = 0.31 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.32 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {15 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{4 \, a \mathrm {sgn}\left (a x + 1\right )} - \frac {c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3}}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {135 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{3} {\left | a \right |} + 360 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{3} + 150 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} c^{3} {\left | a \right |} + 720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{3} + 1120 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{3} - 150 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} + 560 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} - 135 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} + 184 \, a c^{3}}{60 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{5} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 4.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {\frac {7\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {61\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {43\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{30}+\frac {827\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{30}+\frac {269\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{12}+\frac {23\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{4}}{a+\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {5\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}}-\frac {15\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}+\frac {2\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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