Integrand size = 18, antiderivative size = 152 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {315 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
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Time = 0.31 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6310, 6313, 1819, 1821, 821, 272, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {315 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}-\frac {67}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+30 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+2 a^2 c^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{4} a^3 c^3 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1819
Rule 1821
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left (\left (a^3 c^3\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^3 x^3 \, dx\right ) \\ & = \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^6}{x^5 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\left (a^3 c^3\right ) \text {Subst}\left (\int \frac {-1+\frac {6 x}{a}-\frac {16 x^2}{a^2}+\frac {26 x^3}{a^3}-\frac {31 x^4}{a^4}}{x^5 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{4} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {-\frac {24}{a}+\frac {67 x}{a^2}-\frac {104 x^2}{a^3}+\frac {124 x^3}{a^4}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{12} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {-\frac {201}{a^2}+\frac {360 x}{a^3}-\frac {372 x^2}{a^4}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{24} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {-\frac {720}{a^3}+\frac {945 x}{a^4}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {\left (315 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a} \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {\left (315 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a} \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{8} \left (315 a c^3\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = \frac {32 c^3 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+30 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {67}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+2 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {315 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.57 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {1}{8} c^3 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (496+173 a x-51 a^2 x^2+14 a^3 x^3-2 a^4 x^4\right )}{1+a x}-\frac {315 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+67 a x -240\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{8 a}-\frac {\left (\frac {315 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{8 \sqrt {a^{2}}}-\frac {32 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(157\) |
default | \(-\frac {\left (2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+69 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-16 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +138 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-69 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-32 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -384 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+384 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+69 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -138 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +112 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-768 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +768 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -69 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -384 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+384 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{8 a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(542\) |
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Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.75 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, a} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=- c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]
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Time = 0.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.61 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{8} \, {\left (\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {256 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} - \frac {2 \, {\left (325 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 765 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 643 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 187 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - a^{2}}\right )} a \]
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\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\int { -{\left (a c x - c\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.31 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {\frac {187\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {643\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {765\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{4}-\frac {325\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a-\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {6\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {32\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {315\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
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