Integrand size = 22, antiderivative size = 138 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]
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Time = 0.28 (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} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4}-\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4} \]
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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 )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^3}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^7} \\ & = -\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-5 c^3-\frac {15 c^3 x}{a}-\frac {16 c^3 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 {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 c^3+\frac {45 c^3 x}{a}+\frac {42 c^3 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 {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-15 c^3-\frac {45 c^3 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 c^7} \\ & = -\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^4} \\ & = -\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c^4} \\ & = -\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {(3 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^4} \\ & = -\frac {4 \left (a+\frac {1}{x}\right )}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {5 a+\frac {7}{x}}{5 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {15 a+\frac {19}{x}}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {-24+33 a x+18 a^2 x^2-34 a^3 x^3+5 a^4 x^4+15 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{5 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \]
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Time = 0.24 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.63
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {6 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {24 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{6} \left (x -\frac {1}{a}\right )}\right ) a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x -1\right )}\) | \(225\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+85 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -750 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{40 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c^{4} \left (a x -1\right )^{4} \sqrt {a^{2}}}\) | \(436\) |
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Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (5 \, a^{4} x^{4} - 34 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 33 \, a x - 24\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \int \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 4 a^{3} x^{3} + 6 a^{2} x^{2} - 4 a x + 1}\, dx}{c^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{20} \, a {\left (\frac {\frac {9 \, {\left (a x - 1\right )}}{a x + 1} + \frac {75 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {125 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c^{4} {\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c^{4}} \]
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Time = 3.87 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {15\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {25\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {9\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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