Integrand size = 22, antiderivative size = 269 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{8 a}-\frac {3 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
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Time = 0.15 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6329, 99, 159, 163, 41, 222, 94, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {3 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a}+c^3 x \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{9/2}+\frac {6 c^3 \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}}{5 a}+\frac {27 c^3 \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}}{20 a}+\frac {5 c^3 \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {3 c^3 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {1-\frac {1}{a x}}}{8 a}+\frac {21 c^3 \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{8 a}+\frac {3 c^3 \csc ^{-1}(a x)}{8 a} \]
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Rule 41
Rule 94
Rule 99
Rule 159
Rule 163
Rule 214
Rule 222
Rule 6329
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x-c^3 \text {Subst}\left (\int \frac {\left (-\frac {3}{a}-\frac {6 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{7/2} \sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {1}{5} \left (a c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {15}{a^2}-\frac {27 x}{a^3}\right ) \left (1-\frac {x}{a}\right )^{5/2} \sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {1}{20} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {60}{a^3}-\frac {75 x}{a^4}\right ) \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {1}{60} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {180}{a^4}-\frac {45 x}{a^5}\right ) \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x-\frac {1}{120} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {360}{a^5}+\frac {315 x}{a^6}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {21 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {1}{120} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {\frac {360}{a^6}+\frac {45 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {21 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}+\frac {\left (3 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 )}{a} \\ & = \frac {21 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}-\frac {\left (3 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 )}{a^2} \\ & = \frac {21 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{8 a}-\frac {3 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.41 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (-8+30 a x-24 a^2 x^2-55 a^3 x^3+152 a^4 x^4+40 a^5 x^5\right )+15 a^4 x^4 \arcsin \left (\frac {1}{a x}\right )-120 a^4 x^4 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{40 a^5 x^4} \]
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Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {\left (a x +1\right ) \left (152 a^{4} x^{4}-55 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -8\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{40 x^{5} a^{6}}+\frac {\left (-\frac {3 a^{6} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {3 a^{5} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}+a^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{6} \left (a x -1\right )}\) | \(173\) |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{3} \left (-120 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+120 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-15 a^{5} x^{5} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}-15 a^{5} x^{5} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-25 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-32 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -8 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{40 \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{5} \sqrt {a^{2}}}\) | \(281\) |
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Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {30 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (40 \, a^{6} c^{3} x^{6} + 192 \, a^{5} c^{3} x^{5} + 97 \, a^{4} c^{3} x^{4} - 79 \, a^{3} c^{3} x^{3} + 6 \, a^{2} c^{3} x^{2} + 22 \, a c^{3} x - 8 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a^{6} x^{5}} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{7} + x^{6}}\, dx + \int \left (- \frac {a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{6} + x^{5}}\right )\, dx + \int \left (- \frac {3 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {3 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \frac {3 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\, dx + \int \left (- \frac {3 a^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\right )\, dx + \int \left (- \frac {a^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{7} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.12 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {1}{20} \, {\left (\frac {15 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {105 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 465 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 298 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 842 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 575 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 135 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {5 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac {4 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \]
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Time = 0.30 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.47 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {3 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{4 \, a} + \frac {3 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {55 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 200 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 10 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{3} \mathrm {sgn}\left (a x + 1\right ) + 800 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{3} \mathrm {sgn}\left (a x + 1\right ) + 10 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 560 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 55 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 152 \, a c^{3} \mathrm {sgn}\left (a x + 1\right )}{20 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{5} a {\left | a \right |}} \]
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Time = 0.11 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {\frac {27\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {115\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {421\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{10}+\frac {149\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{10}-\frac {93\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}-\frac {21\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{4}}{a+\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {5\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}}-\frac {3\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}-\frac {6\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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