Integrand size = 24, antiderivative size = 335 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {5 \left (304 a-\frac {65}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {135 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {65 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {15 \left (c-\frac {c}{a x}\right )^{9/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{9/2}} \]
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Time = 0.20 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6317, 6314, 100, 155, 158, 152, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {x \left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {65 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {135 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {5 \left (304 a-\frac {65}{x}\right ) \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {15 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \left (c-\frac {c}{a x}\right )^{9/2}}{a \left (1-\frac {1}{a x}\right )^{9/2}} \]
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Rule 65
Rule 100
Rule 152
Rule 155
Rule 158
Rule 214
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-\frac {c}{a x}\right )^{9/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{9/2} \, dx}{\left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {\left (c-\frac {c}{a x}\right )^{9/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^6}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (c-\frac {c}{a x}\right )^{9/2} \text {Subst}\left (\int \frac {\left (\frac {15}{2 a}+\frac {5 x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )^4}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 a \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {15}{4 a^2}-\frac {65 x}{4 a^3}\right ) \left (1-\frac {x}{a}\right )^3}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {65 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (4 a^2 \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {105}{8 a^3}-\frac {675 x}{8 a^4}\right ) \left (1-\frac {x}{a}\right )^2}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{7 \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {135 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {65 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (8 a^3 \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {525}{16 a^4}-\frac {4875 x}{16 a^5}\right ) \left (1-\frac {x}{a}\right )}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{35 \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {5 \left (304 a-\frac {65}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {135 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {65 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (15 \left (c-\frac {c}{a x}\right )^{9/2}\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 )^{9/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {5 \left (304 a-\frac {65}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {135 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {65 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (15 \left (c-\frac {c}{a x}\right )^{9/2}\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 )^{9/2}} \\ & = \frac {10 \left (a-\frac {1}{x}\right )^4 \left (c-\frac {c}{a x}\right )^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {5 \left (304 a-\frac {65}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {135 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {65 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^5 \left (c-\frac {c}{a x}\right )^{9/2} x}{a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {15 \left (c-\frac {c}{a x}\right )^{9/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{9/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.42 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (-2+20 a x-110 a^2 x^2+720 a^3 x^3+1685 a^4 x^4+7 a^5 x^5-35 a^4 \sqrt {1+\frac {1}{a x}} x^4 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )+70 a^4 x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {1}{a x}\right )\right )}{7 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4} \]
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Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (14 \sqrt {\left (a x +1\right ) x}\, a^{\frac {11}{2}} x^{5}+3510 a^{\frac {9}{2}} \sqrt {\left (a x +1\right ) x}\, x^{4}+1440 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}-105 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{5} x^{5}-105 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{4} x^{4}-220 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}+40 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{14 \left (a x -1\right )^{2} x^{3} a^{\frac {9}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(229\) |
risch | \(\frac {\left (7 a^{5} x^{5}+859 a^{4} x^{4}+720 a^{3} x^{3}-110 a^{2} x^{2}+20 a x -2\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{7 x^{3} a^{4} \left (a x -1\right )}+\frac {\left (-\frac {15 a^{4} \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}+\frac {128 a^{2} \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{c \left (x +\frac {1}{a}\right )}\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a^{4} \left (a x -1\right )}\) | \(231\) |
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Time = 0.30 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.30 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {105 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{28 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac {105 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{14 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Timed out} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int {\left (c-\frac {c}{a\,x}\right )}^{9/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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