Integrand size = 20, antiderivative size = 62 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {8 c \sqrt {1-\frac {1}{a^2 x^2}} x}{3 \sqrt {c-a c x}}+\frac {2}{3} \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x} \]
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Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6311, 6316, 79, 37} \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {10 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]
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Rule 37
Rule 79
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{-\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 {1-\frac {x}{a}}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {2 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {\left (5 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 a \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {10 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} (-5+a x) \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -5\right ) \left (a x +1\right )}{3 \sqrt {-c \left (a x -1\right )}\, a}\) | \(42\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (a x -5\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{3 \left (a x -1\right ) a}\) | \(47\) |
default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a x -5\right )}{3 \left (a x -1\right ) a}\) | \(48\) |
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Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{2} \sqrt {-c} x^{2} - 4 \, a \sqrt {-c} x - 5 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{3 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.69 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (-a c x - c\right )}^{\frac {3}{2}} {\left | c \right |}}{3 \, a c^{2}} + \frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a c} \]
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Time = 0.00 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\left (a\,x-3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a}-\frac {16\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]
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