Integrand size = 24, antiderivative size = 118 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}} \]
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Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6268, 25, 528, 382, 79, 53, 65, 214} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {9 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}-\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}} \]
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Rule 25
Rule 53
Rule 65
Rule 79
Rule 214
Rule 382
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx \\ & = -\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{5/2} (1-a x)} \, dx \\ & = \frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{7/2} x} \, dx}{a} \\ & = \frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {(9 c) \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a c} \\ & = -\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2} \\ & = -\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^3} \\ & = -\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.49 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1-\frac {1}{a x}\right )}{5 a \left (c-\frac {c}{a x}\right )^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(102)=204\).
Time = 0.54 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.09
method | result | size |
risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {9 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {4 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{5 a^{7} c \left (x -\frac {1}{a}\right )^{3}}-\frac {18 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{5 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {54 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+\left (x -\frac {1}{a}\right ) a c}}{5 a^{5} c \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {c \left (a x -1\right ) a x}}{c^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(247\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (90 a^{\frac {9}{2}} \sqrt {\left (a x -1\right ) x}\, x^{4}+45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{4}-80 a^{\frac {7}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{2}-360 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}-180 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{3}+132 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x +540 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}+270 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{2}-60 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}-360 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} x -180 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a x +90 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{10 \sqrt {\left (a x -1\right ) x}\, c^{3} \sqrt {a}\, \left (a x -1\right )^{4}}\) | \(328\) |
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Time = 0.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.49 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {45 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, -\frac {45 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x - 1\right )}\, dx \]
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\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (102) = 204\).
Time = 0.48 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.68 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {9 \, \log \left (c^{4} {\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{14 \, a c^{\frac {5}{2}}} - \frac {9 \, \log \left ({\left | 2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} \sqrt {c} {\left | a \right |} - 13 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a c + 36 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} c^{\frac {3}{2}} {\left | a \right |} - 55 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a c^{2} + 50 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} c^{\frac {5}{2}} {\left | a \right |} - 27 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a c^{3} + 8 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} c^{\frac {7}{2}} {\left | a \right |} - a c^{4} \right |}\right ) \mathrm {sgn}\left (x\right )}{14 \, a c^{\frac {5}{2}}} + \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |} \mathrm {sgn}\left (x\right )}{a^{2} c^{3}} \]
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Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a\,x-1\right )} \,d x \]
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