Integrand size = 27, antiderivative size = 163 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=4 a^4 \sqrt {c-\frac {c}{a x}}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6268, 25, 528, 457, 90, 52, 65, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 a^4 \sqrt {c-\frac {c}{a x}} \]
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Rule 25
Rule 52
Rule 65
Rule 90
Rule 214
Rule 457
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^5 (1+a x)} \, dx \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x^4 (1+a x)} \, dx}{c} \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^5} \, dx}{c} \\ & = -\frac {a \text {Subst}\left (\int \frac {x^3 \left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {a \text {Subst}\left (\int \left (a^2 \left (c-\frac {c x}{a}\right )^{3/2}-\frac {a^3 \left (c-\frac {c x}{a}\right )^{3/2}}{a+x}-\frac {a^2 \left (c-\frac {c x}{a}\right )^{5/2}}{c}+\frac {a^2 \left (c-\frac {c x}{a}\right )^{7/2}}{c^2}\right ) \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\frac {a^4 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\left (2 a^4\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right ) \\ & = 4 a^4 \sqrt {c-\frac {c}{a x}}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\left (4 a^4 c\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = 4 a^4 \sqrt {c-\frac {c}{a x}}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}-\left (8 a^5\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right ) \\ & = 4 a^4 \sqrt {c-\frac {c}{a x}}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (35-95 a x+138 a^2 x^2-236 a^3 x^3+788 a^4 x^4\right )}{315 x^4}-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {2 \left (788 a^{5} x^{5}-1024 a^{4} x^{4}+374 a^{3} x^{3}-233 a^{2} x^{2}+130 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \left (a x -1\right )}+\frac {2 a^{4} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{\sqrt {c}\, \left (a x -1\right )}\) | \(168\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (630 \sqrt {\left (a x -1\right ) x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{6}-1890 \sqrt {a \,x^{2}-x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{6}+1260 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{4}+945 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{6}-630 a^{\frac {9}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{6}-945 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{5} x^{6}-316 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{3}+156 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-120 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+70 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{315 x^{5} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(326\) |
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Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}, \frac {2 \, {\left (630 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (136) = 272\).
Time = 0.88 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.66 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {4 \, \sqrt {2} a^{5} c \arctan \left (-\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a^{9} c - 1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} a^{8} c^{\frac {3}{2}} {\left | a \right |} + 2100 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a^{9} c^{2} - 3150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{8} c^{\frac {5}{2}} {\left | a \right |} + 3528 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{9} c^{3} - 2625 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{8} c^{\frac {7}{2}} {\left | a \right |} + 1215 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{9} c^{4} - 315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{8} c^{\frac {9}{2}} {\left | a \right |} + 35 \, a^{9} c^{5}\right )}}{315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{x^5\,\left (a\,x+1\right )} \,d x \]
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