Integrand size = 22, antiderivative size = 138 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \]
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Time = 0.35 (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} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2}-\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac {c x}{a}\right )^5} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^5}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^7} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-5 c^5-\frac {25 c^5 x}{a}-\frac {39 c^5 x^2}{a^2}+\frac {5 c^5 x^3}{a^3}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^7} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 c^5+\frac {75 c^5 x}{a}+\frac {88 c^5 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 {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-15 c^5-\frac {75 c^5 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 c^7} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {5 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^2} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {5 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c^2} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {(5 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^2} \\ & = -\frac {16 \left (a+\frac {1}{x}\right )}{5 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {11}{x}\right )}{15 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {75 a+\frac {103}{x}}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}+\frac {5 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {-118+161 a x+91 a^2 x^2-173 a^3 x^3+15 a^4 x^4+75 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \]
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Time = 0.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.63
method | result | size |
risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{6} \left (x -\frac {1}{a}\right )^{3}}-\frac {52 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{15 a^{5} \left (x -\frac {1}{a}\right )^{2}}-\frac {143 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{15 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(225\) |
default | \(-\frac {-75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-75 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+60 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+300 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+300 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-97 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -450 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-450 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+43 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+300 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +300 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -75 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-75 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{15 a \left (a x -1\right )^{2} \sqrt {a^{2}}\, c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(438\) |
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Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {75 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 75 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 173 \, a^{3} x^{3} + 91 \, a^{2} x^{2} + 161 \, a x - 118\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a^{2} \int \frac {x^{2}}{\frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {1}{15} \, a {\left (\frac {\frac {17 \, {\left (a x - 1\right )}}{a x + 1} + \frac {100 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {150 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {75 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {75 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]
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Time = 0.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {5 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{2} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{2} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.87 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {10\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {20\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {10\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {17\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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