Integrand size = 24, antiderivative size = 219 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {10 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (112 a-\frac {29}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {11 \left (c-\frac {c}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{5/2}} \]
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Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6317, 6314, 100, 155, 152, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {x \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {10 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {\left (112 a-\frac {29}{x}\right ) \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {11 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}{a \left (1-\frac {1}{a x}\right )^{5/2}} \]
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Rule 65
Rule 100
Rule 152
Rule 155
Rule 214
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-\frac {c}{a x}\right )^{5/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = -\frac {\left (c-\frac {c}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (c-\frac {c}{a x}\right )^{5/2} \text {Subst}\left (\int \frac {\left (\frac {11}{2 a}+\frac {x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )^2}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 a \left (c-\frac {c}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {11}{4 a^2}-\frac {29 x}{4 a^3}\right ) \left (1-\frac {x}{a}\right )}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (112 a-\frac {29}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (11 \left (c-\frac {c}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (112 a-\frac {29}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (11 \left (c-\frac {c}{a x}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (112 a-\frac {29}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{5/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {11 \left (c-\frac {c}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.57 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (-2+32 a x+103 a^2 x^2+3 a^3 x^3-3 a^2 \sqrt {1+\frac {1}{a x}} x^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )+30 a^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {1}{a x}\right )\right )}{3 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2} \]
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Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (6 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}+266 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-33 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} x^{3}+64 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-33 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}-4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{6 \left (a x -1\right )^{2} x \,a^{\frac {5}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(195\) |
risch | \(\frac {\left (3 a^{3} x^{3}+37 a^{2} x^{2}+32 a x -2\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \,a^{2} \left (a x -1\right )}+\frac {\left (-\frac {11 a^{2} \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}+\frac {32 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{c \left (x +\frac {1}{a}\right )}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a^{2} \left (a x -1\right )}\) | \(212\) |
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Time = 0.29 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.74 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\left [\frac {33 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{3} c^{2} x^{3} + 133 \, a^{2} c^{2} x^{2} + 32 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, \frac {33 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (3 \, a^{3} c^{2} x^{3} + 133 \, a^{2} c^{2} x^{2} + 32 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Timed out} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int {\left (c-\frac {c}{a\,x}\right )}^{5/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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