Integrand size = 18, antiderivative size = 125 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {152 \left (a+\frac {1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}} \]
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Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6310, 6313, 866, 1649, 803, 665} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {152 \left (a+\frac {1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Rule 665
Rule 803
Rule 866
Rule 1649
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^5 x^5} \, dx}{a^5 c^5} \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^8} \, dx,x,\frac {1}{x}\right )}{a^5 c^5} \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (1+\frac {x}{a}\right )^8}{\left (1-\frac {x^2}{a^2}\right )^{13/2}} \, dx,x,\frac {1}{x}\right )}{a^5 c^5} \\ & = \frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^7 \left (8 a^3+11 a^2 x+11 a x^2\right )}{\left (1-\frac {x^2}{a^2}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{11 a^5 c^5} \\ & = -\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}+\frac {\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^6 \left (138 a^3+99 a^2 x\right )}{\left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{99 a^5 c^5} \\ & = \frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {152 \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^5}{\left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{231 a^2 c^5} \\ & = -\frac {152 \left (a+\frac {1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (1+a x)^2 \left (-152+61 a x-16 a^2 x^2+2 a^3 x^3\right )}{1155 c^5 (-1+a x)^6} \]
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Time = 0.43 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x +1\right )}{1155 \left (a x -1\right )^{4} c^{5} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(58\) |
default | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x +1\right )}{1155 \left (a x -1\right )^{4} c^{5} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(58\) |
trager | \(-\frac {\left (a x +1\right ) \left (2 a^{5} x^{5}-12 a^{4} x^{4}+31 a^{3} x^{3}-46 a^{2} x^{2}-243 a x -152\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{1155 a \,c^{5} \left (a x -1\right )^{6}}\) | \(76\) |
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Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {{\left (2 \, a^{6} x^{6} - 10 \, a^{5} x^{5} + 19 \, a^{4} x^{4} - 15 \, a^{3} x^{3} - 289 \, a^{2} x^{2} - 395 \, a x - 152\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1155 \, {\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, a^{2} c^{5} x + a c^{5}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=- \frac {\int \frac {1}{\frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {6 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {15 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {20 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {15 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {6 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\frac {385 \, {\left (a x - 1\right )}}{a x + 1} - \frac {495 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {231 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 105}{9240 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}}} \]
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Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.32 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {4 \, {\left (1155 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{7} x^{7} + 2079 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{6} x^{6} + 2541 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{5} x^{5} + 825 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{4} x^{4} + 165 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} - 55 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 11 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{1155 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{11} a c^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.58 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {\frac {3\,{\left (a\,x-1\right )}^2}{7\,{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{5\,{\left (a\,x+1\right )}^3}-\frac {a\,x-1}{3\,\left (a\,x+1\right )}+\frac {1}{11}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}} \]
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