Integrand size = 22, antiderivative size = 181 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2} \]
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Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6329, 105, 21, 101, 157, 12, 94, 214} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {24 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}-\frac {9 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {6 \sqrt {\frac {1}{a x}+1}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}} \]
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Rule 12
Rule 21
Rule 94
Rule 101
Rule 105
Rule 157
Rule 214
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c^2} \\ & = \frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{a}-\frac {3 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c^2} \\ & = \frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{a c^2} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {6 \text {Subst}\left (\int \frac {-\frac {5}{2}-\frac {2 x}{a}}{x \left (1-\frac {x}{a}\right )^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a c^2} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {2 \text {Subst}\left (\int \frac {\frac {15}{2 a}+\frac {9 x}{2 a^2}}{x \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 c^2} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {(2 a) \text {Subst}\left (\int -\frac {15}{2 a^2 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 c^2} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {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 c^2} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {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 c^2} \\ & = -\frac {6 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {9 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {24 \sqrt {1+\frac {1}{a x}}}{5 a c^2 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.43 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-24+57 a x-39 a^2 x^2+5 a^3 x^3\right )}{5 (-1+a x)^3}+3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \]
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Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {6 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {24 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{6} \left (x -\frac {1}{a}\right )}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(225\) |
default | \(\frac {120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+125 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-85 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+720 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+148 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +750 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-480 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -67 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-500 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )+125 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{40 a \sqrt {a^{2}}\, \left (a x -1\right )^{2} c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(438\) |
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Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (5 \, a^{4} x^{4} - 34 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 33 \, a x - 24\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4}}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.85 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {1}{20} \, a {\left (\frac {\frac {9 \, {\left (a x - 1\right )}}{a x + 1} + \frac {75 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {125 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.35 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{2} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{2} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 0.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.67 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2}-\frac {\frac {15\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {25\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {9\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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