Integrand size = 27, antiderivative size = 83 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6327, 6328, 45} \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x^m \sqrt {c-a^2 c x^2}}{a (m+1) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^{1+m} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int x^m (-1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (-x^m+a x^{1+m}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\frac {x^{1+m}}{a (1+m)}+\frac {x^{2+m}}{2+m}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.56 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(\frac {x^{1+m} \sqrt {-a^{2} c \,x^{2}+c}\, \left (a m x +a x -m -2\right ) \sqrt {\frac {a x -1}{a x +1}}}{\left (1+m \right ) \left (2+m \right ) \left (a x -1\right )}\) | \(64\) |
risch | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-\frac {c \left (a^{2} x^{2}-1\right )}{\left (a x -1\right ) \left (a x +1\right )}}\, \left (a x +1\right ) c \left (a m x +a x -m -2\right ) x \,x^{m}}{\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-c}\, \left (2+m \right ) \left (1+m \right )}\) | \(97\) |
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Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a m + a\right )} x^{2} - {\left (m + 2\right )} x\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}}{m^{2} - {\left (a m^{2} + 3 \, a m + 2 \, a\right )} x + 3 \, m + 2} \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (a \sqrt {-c} {\left (m + 1\right )} x^{2} - \sqrt {-c} {\left (m + 2\right )} x\right )} {\left (a x - 1\right )} x^{m}}{{\left (m^{2} + 3 \, m + 2\right )} a x - m^{2} - 3 \, m - 2} \]
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\[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Time = 4.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.13 \[ \int e^{-\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (\frac {x^m\,x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (m+1\right )}{m^2+3\,m+2}-\frac {x\,x^m\,\sqrt {c-a^2\,c\,x^2}\,\left (m+2\right )}{a\,\left (m^2+3\,m+2\right )}\right )}{x-\frac {1}{a}} \]
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