Integrand size = 27, antiderivative size = 113 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-4 a^3 \sqrt {c-\frac {c}{a x}}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 113, 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} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt {c-\frac {c}{a x}} \]
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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^4} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^4 (1+a x)} \, dx \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x^3 (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^4} \, dx}{c} \\ & = -\frac {a \text {Subst}\left (\int \frac {x^2 \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 (\frac {a^2 \left (c-\frac {c x}{a}\right )^{3/2}}{a+x}-\frac {a \left (c-\frac {c x}{a}\right )^{5/2}}{c}\right ) \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {a^3 \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^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\left (2 a^3\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right ) \\ & = -4 a^3 \sqrt {c-\frac {c}{a x}}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\left (4 a^3 c\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -4 a^3 \sqrt {c-\frac {c}{a x}}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\left (8 a^4\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^3 \sqrt {c-\frac {c}{a x}}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (3-9 a x+16 a^2 x^2-52 a^3 x^3\right )}{21 x^3}+4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Time = 0.51 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42
method | result | size |
risch | \(-\frac {2 \left (52 a^{4} x^{4}-68 a^{3} x^{3}+25 a^{2} x^{2}-12 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{21 x^{3} \left (a x -1\right )}-\frac {2 a^{3} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{\sqrt {c}\, \left (a x -1\right )}\) | \(160\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (42 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{5}-126 \sqrt {a \,x^{2}-x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{5}+84 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{3}+63 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{5}-42 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{5}-63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{4} x^{5}-20 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+12 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}-6 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{21 x^{4} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(302\) |
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Time = 0.26 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.78 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\left [\frac {2 \, {\left (21 \, \sqrt {2} a^{3} \sqrt {c} x^{3} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{21 \, x^{3}}, -\frac {2 \, {\left (42 \, \sqrt {2} a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{21 \, x^{3}}\right ] \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{4} \left (a x + 1\right )}\, dx \]
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\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (94) = 188\).
Time = 0.74 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.15 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {4 \, \sqrt {2} a^{4} c \arctan \left (-\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (84 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a^{7} c - 84 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{6} c^{\frac {3}{2}} {\left | a \right |} + 112 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{7} c^{2} - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{6} c^{\frac {5}{2}} {\left | a \right |} + 63 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{7} c^{3} - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{6} c^{\frac {7}{2}} {\left | a \right |} + 3 \, a^{7} c^{4}\right )}}{21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} a^{3} {\left | a \right |} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{x^4\,\left (a\,x+1\right )} \,d x \]
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