Integrand size = 22, antiderivative size = 313 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {5}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {5 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \]
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Time = 0.19 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6326, 6330, 96, 94, 214} \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {1}{6} a^5 c^3 x^6 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {1}{6} a^4 c^3 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {1}{8} a^3 c^3 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {1}{24} a^2 c^3 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {5 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{16 a}-\frac {5}{48} a c^3 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {5}{16} c^3 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]
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Rule 94
Rule 96
Rule 214
Rule 6326
Rule 6330
Rubi steps \begin{align*} \text {integral}& = -\left (\left (a^6 c^3\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^3 x^6 \, dx\right ) \\ & = \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^8} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\left (a^5 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^7} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7+\frac {1}{6} \left (5 a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^6} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {1}{2} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7+\frac {1}{8} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/2}}{x^4 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7+\frac {1}{24} \left (5 a c^3\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7+\frac {1}{16} \left (5 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a} \\ & = -\frac {5}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a^2} \\ & = -\frac {5}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {5 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (48+231 a x-144 a^2 x^2-182 a^3 x^3+144 a^4 x^4+56 a^5 x^5-48 a^6 x^6\right )-105 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{336 a} \]
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Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {\left (48 a^{6} x^{6}-56 a^{5} x^{5}-144 a^{4} x^{4}+182 a^{3} x^{3}+144 a^{2} x^{2}-231 a x -48\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{336 a}-\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{16 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(144\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{3} \left (48 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-56 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-96 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+126 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -64 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-105 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +112 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+105 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{336 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(231\) |
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Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.47 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (48 \, a^{7} c^{3} x^{7} - 8 \, a^{6} c^{3} x^{6} - 200 \, a^{5} c^{3} x^{5} + 38 \, a^{4} c^{3} x^{4} + 326 \, a^{3} c^{3} x^{3} - 87 \, a^{2} c^{3} x^{2} - 279 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{336 \, a} \]
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\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- c^{3} \left (\int 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- 3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{336} \, {\left (\frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (105 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 700 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 1981 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 3072 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 1981 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 700 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {7 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {21 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {35 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {21 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {7 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \]
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Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.51 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {5 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{16 \, {\left | a \right |}} + \frac {1}{336} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {48 \, c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (231 \, c^{3} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (72 \, a c^{3} \mathrm {sgn}\left (a x + 1\right ) + {\left (91 \, a^{2} c^{3} \mathrm {sgn}\left (a x + 1\right ) - 4 \, {\left (18 \, a^{3} c^{3} \mathrm {sgn}\left (a x + 1\right ) - {\left (6 \, a^{5} c^{3} x \mathrm {sgn}\left (a x + 1\right ) - 7 \, a^{4} c^{3} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Time = 0.12 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {\frac {25\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6}-\frac {5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}-\frac {283\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{24}+\frac {128\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{7}+\frac {283\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{24}-\frac {25\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{6}+\frac {5\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}}{a-\frac {7\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {21\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {35\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {35\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {21\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {7\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}}-\frac {5\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a} \]
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