Integrand size = 22, antiderivative size = 313 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{40} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \]
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Time = 0.20 (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} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} a^6 c^3 x^7 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {1}{14} a^5 c^3 x^6 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {1}{70} a^4 c^3 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {9}{280} a^3 c^3 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {3}{40} a^2 c^3 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{16 a}-\frac {3}{16} a c^3 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {9}{16} c^3 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]
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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^{3 \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 )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^8} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {1}{7} \left (3 a^5 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}}{x^7} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7+\frac {1}{14} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{9/2}}{x^6 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7+\frac {1}{70} \left (9 a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{7/2}}{x^5 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7+\frac {1}{40} \left (9 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 {3}{40} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7+\frac {1}{8} \left (3 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 {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{40} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7+\frac {1}{16} \left (9 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 {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{40} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7+\frac {\left (9 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 {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{40} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {\left (9 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 {9}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {3}{16} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {3}{40} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {9}{280} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{70} a^4 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5+\frac {1}{14} a^5 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {9 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (368-245 a x-656 a^2 x^2-350 a^3 x^3+208 a^4 x^4+280 a^5 x^5+80 a^6 x^6\right )+315 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{560 a} \]
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Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {\left (80 a^{6} x^{6}+280 a^{5} x^{5}+208 a^{4} x^{4}-350 a^{3} x^{3}-656 a^{2} x^{2}-245 a x +368\right ) \left (a x -1\right ) c^{3}}{560 a \sqrt {\frac {a x -1}{a x +1}}}-\frac {9 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{16 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(144\) |
default | \(-\frac {\left (a x -1\right )^{2} c^{3} \left (80 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+280 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+288 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-70 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +192 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-315 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -560 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+315 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{560 a \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(240\) |
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Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.47 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (80 \, a^{7} c^{3} x^{7} + 360 \, a^{6} c^{3} x^{6} + 488 \, a^{5} c^{3} x^{5} - 142 \, a^{4} c^{3} x^{4} - 1006 \, a^{3} c^{3} x^{3} - 901 \, a^{2} c^{3} x^{2} + 123 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{6} x^{6}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{560} \, {\left (\frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (315 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 2100 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 5943 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 9216 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 8393 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2100 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 315 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {7 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {21 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {35 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {21 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {7 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \]
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Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.57 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{16 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {1}{560} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, {\left (\frac {328 \, a c^{3}}{\mathrm {sgn}\left (a x + 1\right )} + {\left (\frac {175 \, a^{2} c^{3}}{\mathrm {sgn}\left (a x + 1\right )} - 4 \, {\left (\frac {26 \, a^{3} c^{3}}{\mathrm {sgn}\left (a x + 1\right )} + 5 \, {\left (\frac {2 \, a^{5} c^{3} x}{\mathrm {sgn}\left (a x + 1\right )} + \frac {7 \, a^{4} c^{3}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x\right )} x\right )} x\right )} x + \frac {245 \, c^{3}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {368 \, c^{3}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} \]
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Time = 4.03 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {\frac {15\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}-\frac {9\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {1199\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}-\frac {1152\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{35}+\frac {849\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{40}-\frac {15\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{2}+\frac {9\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}}{a-\frac {7\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {21\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {35\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {35\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {21\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {7\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}}-\frac {9\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a} \]
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