Integrand size = 18, antiderivative size = 129 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {35}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {35 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
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Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6310, 6313, 1819, 1821, 821, 272, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {35 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {5}{2} a c^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {35}{3} c^2 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {1}{3} a^2 c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}} \]
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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 (a^2 c^2\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^2 x^2 \, dx \\ & = -\left (\left (a^2 c^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^5}{x^4 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\left (a^2 c^2\right ) \text {Subst}\left (\int \frac {-1+\frac {5 x}{a}-\frac {11 x^2}{a^2}+\frac {15 x^3}{a^3}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{3} \left (a^2 c^2\right ) \text {Subst}\left (\int \frac {-\frac {15}{a}+\frac {35 x}{a^2}-\frac {45 x^2}{a^3}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {1}{6} \left (a^2 c^2\right ) \text {Subst}\left (\int \frac {-\frac {70}{a^2}+\frac {105 x}{a^3}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {35}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {\left (35 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {35}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3+\frac {\left (35 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a} \\ & = \frac {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {35}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{2} \left (35 a c^2\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 {16 c^2 \left (a-\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {35}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {5}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {35 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {1}{6} c^2 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (166+55 a x-13 a^2 x^2+2 a^3 x^3\right )}{1+a x}-\frac {105 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-15 a x +70\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{6 a}+\frac {\left (-\frac {35 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{2 \sqrt {a^{2}}}+\frac {16 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(148\) |
default | \(-\frac {\left (15 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+30 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -120 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+15 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +46 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-240 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -120 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+120 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) c^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 a \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(474\) |
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Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.81 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx + \int \left (- \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right ) \]
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Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.58 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {1}{6} \, a {\left (\frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {96 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}} + \frac {2 \, {\left (87 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 136 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 57 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\int { {\left (a c x - c\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Time = 4.54 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.26 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {19\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {136\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+29\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {16\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {35\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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