Integrand size = 22, antiderivative size = 103 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}-\frac {c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6312, 827, 829, 858, 222, 272, 65, 214} \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right )}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a} \]
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 827
Rule 829
Rule 858
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right ) \left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {1}{2} c^3 \text {Subst}\left (\int \frac {\left (\frac {2 c}{a}+\frac {6 c x}{a^2}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}-\frac {1}{4} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {-\frac {4 c}{a^3}-\frac {6 c x}{a^4}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {\left (3 c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {c^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}+\frac {c^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a} \\ & = \frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}-\left (a c^4\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 {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}-\frac {c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.70 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {c^4 \left (-8+12 a x+40 a^2 x^2+12 a^3 x^3-32 a^4 x^4-24 a^5 x^5+42 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-15 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \arcsin \left (\frac {1}{a x}\right )+24 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{24 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4} \]
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Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\frac {\left (a x -1\right ) \left (8 a^{2} x^{2}+3 a x -2\right ) c^{4}}{6 x^{3} a^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {a^{4} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {3 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}+a^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{4} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(157\) |
default | \(-\frac {\left (a x -1\right )^{2} c^{4} \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-9 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}}\) | \(233\) |
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Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.51 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {18 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{4} x^{4} + 14 \, a^{3} c^{4} x^{3} + 11 \, a^{2} c^{4} x^{2} + a c^{4} x - 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (\int \left (- \frac {4 a}{\frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{2}}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{3}}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (91) = 182\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.17 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {1}{3} \, {\left (\frac {9 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {3 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 29 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (91) = 182\).
Time = 0.30 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.41 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {3 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4}}{a \mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} - 12 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} - 3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} - 8 \, a c^{4}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 0.14 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.78 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {5\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {29\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}+c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {3\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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