Integrand size = 22, antiderivative size = 138 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5} \]
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Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5}-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^5} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (c-\frac {c x}{a}\right )^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = -\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^2}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^7} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-5 c^2-\frac {10 c^2 x}{a}-\frac {8 c^2 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^7} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 c^2+\frac {30 c^2 x}{a}+\frac {26 c^2 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{15 c^7} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-15 c^2-\frac {30 c^2 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 c^7} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^5} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c^5} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^5} \\ & = -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {-56+82 a x+32 a^2 x^2-76 a^3 x^3+15 a^4 x^4+30 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(122)=244\).
Time = 0.24 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.88
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{5}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{5} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{10 a^{9} \left (x -\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{60 a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {383 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{120 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{8 a^{7} \left (x +\frac {1}{a}\right )}\right ) a^{5} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{5} \left (a x -1\right )}\) | \(259\) |
default | \(-\frac {\left (-75 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}+45 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+150 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-64 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-300 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-14 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+37 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+150 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-60 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{30 a \sqrt {a^{2}}\, c^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{5}}\) | \(615\) |
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Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {a^{5} \left (\int \left (- \frac {x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\right )\, dx + \int \frac {a x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\, dx\right )}{c^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {1}{120} \, a {\left (\frac {\frac {32 \, {\left (a x - 1\right )}}{a x + 1} + \frac {310 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {585 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{5}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{5}} + \frac {15 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{5}}\right )} \]
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Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^5}-\frac {\frac {62\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {39\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {32\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}+\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^5} \]
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