Integrand size = 22, antiderivative size = 105 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c}-\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c} \]
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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 )^4} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^4}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c^5} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-3 c^4-\frac {12 c^4 x}{a}-\frac {13 c^4 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^5} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\text {Subst}\left (\int \frac {3 c^4+\frac {12 c^4 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 c^5} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {(4 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} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {4 \left (3 a+\frac {4}{x}\right )}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}+\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (19-26 a x+3 a^2 x^2\right )}{(-1+a x)^2}+12 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{3 a c} \]
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Time = 0.16 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.74
method | result | size |
risch | \(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {4 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{3 a^{4} \left (x -\frac {1}{a}\right )^{2}}-\frac {20 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{3 a^{3} \left (x -\frac {1}{a}\right )}\right ) a \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(183\) |
default | \(\frac {12 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-36 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-9 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -36 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+36 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +7 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+36 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -12 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{3 a \left (a x -1\right ) \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(346\) |
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Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.22 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - 7 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \int \frac {x}{\frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \]
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Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.27 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2}{3} \, a {\left (\frac {\frac {8 \, {\left (a x - 1\right )}}{a x + 1} - \frac {12 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {6 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.60 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {4 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 3.76 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c}-\frac {\frac {16\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {8\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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