Integrand size = 20, antiderivative size = 138 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3}-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3} \]
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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 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^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 )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^7} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-5 c^4-\frac {20 c^4 x}{a}-\frac {27 c^4 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 {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 c^4+\frac {60 c^4 x}{a}+\frac {64 c^4 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 {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-15 c^4-\frac {60 c^4 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 c^7} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^3} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c^3} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\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^3} \\ & = -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {-94+128 a x+73 a^2 x^2-134 a^3 x^3+15 a^4 x^4+60 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \]
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Time = 0.17 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.63
method | result | size |
risch | \(\frac {a x -1}{a \,c^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {4 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{3} \sqrt {a^{2}}}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{5 a^{7} \left (x -\frac {1}{a}\right )^{3}}-\frac {31 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{15 a^{6} \left (x -\frac {1}{a}\right )^{2}}-\frac {104 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{15 a^{5} \left (x -\frac {1}{a}\right )}\right ) a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(225\) |
default | \(-\frac {-60 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+45 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+240 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-76 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -360 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-360 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+34 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+240 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -60 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-60 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{15 a \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\) | \(431\) |
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Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {60 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 60 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 134 \, a^{3} x^{3} + 73 \, a^{2} x^{2} + 128 \, a x - 94\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^{3} \int \frac {x^{3}}{a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {1}{30} \, a {\left (\frac {\frac {22 \, {\left (a x - 1\right )}}{a x + 1} + \frac {155 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {240 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {120 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {120 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.46 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {4 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{3} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 3.81 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {31\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {16\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {22\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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