Integrand size = 18, antiderivative size = 100 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {11}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {5 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
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Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6310, 6313, 1821, 821, 272, 65, 214} \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {5 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {3}{2} a c^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {11}{3} c^2 x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {1}{3} a^2 c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int e^{-\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 )^3}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \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 {9}{a}-\frac {11 x}{a^2}+\frac {3 x^2}{a^3}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{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 {22}{a^2}-\frac {15 x}{a^3}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {11}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3}{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 (5 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 {11}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3}{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 (5 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 {11}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3}{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 (5 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 {11}{3} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3}{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 {5 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.64 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (22-9 a x+2 a^2 x^2\right )-15 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \]
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Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-9 a x +22\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{6 a}-\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(112\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -24 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}}\) | \(176\) |
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Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c^{2} x^{3} - 7 \, a^{2} c^{2} x^{2} + 13 \, a c^{2} x + 22 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]
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\[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \left (- 2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (84) = 168\).
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.81 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=-\frac {1}{6} \, a {\left (\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {2 \, {\left (33 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 40 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {5 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{2 \, {\left | a \right |}} + \frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, a c^{2} x \mathrm {sgn}\left (a x + 1\right ) - 9 \, c^{2} \mathrm {sgn}\left (a x + 1\right )\right )} x + \frac {22 \, c^{2} \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]
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Time = 4.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.40 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx=\frac {5\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}-\frac {40\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+11\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}-\frac {5\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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