Integrand size = 20, antiderivative size = 137 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {46 \sqrt {c-a c x}}{3 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}} \]
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Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 91, 79, 37} \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {46 \sqrt {c-a c x}}{3 a^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rule 37
Rule 79
Rule 91
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \sqrt {x} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (23 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a^2 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {20 \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {46 \sqrt {c-a c x}}{3 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}+\frac {2 x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (-23-10 a x+a^2 x^2\right )}{3 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.46 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.40
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-10 a x -23\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a \left (a x -1\right )^{2}}\) | \(55\) |
default | \(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a^{2} x^{2}-10 a x -23\right )}{3 \left (a x -1\right )^{2} a}\) | \(56\) |
risch | \(-\frac {2 \left (a x -11\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{3 a \sqrt {-c \left (a x -1\right )}}+\frac {8 c \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(74\) |
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Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.36 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} x^{2} - 10 \, a x - 23\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.55 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} \sqrt {-c} x^{3} - 9 \, a^{2} \sqrt {-c} x^{2} - 33 \, a \sqrt {-c} x - 23 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{3 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.52 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\left (a\,x-9\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a}-\frac {64\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]
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