Integrand size = 25, antiderivative size = 80 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^m}{a m \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^{1+m}}{(1+m) \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6332, 6328, 45} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\frac {x^{m+1} \sqrt {c-\frac {c}{a^2 x^2}}}{(m+1) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^m \sqrt {c-\frac {c}{a^2 x^2}}}{a m \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 45
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^m \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int x^{-1+m} (1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (x^{-1+m}+a x^m\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x^m}{a m \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^{1+m}}{(1+m) \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.65 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {x^m}{a m}+\frac {x^{1+m}}{1+m}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79
method | result | size |
gosper | \(\frac {x^{1+m} \left (a m x +m +1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{m \left (1+m \right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(63\) |
risch | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{\left (a x -1\right ) \left (a x +1\right )}}\, \left (a x -1\right ) \left (a m x +m +1\right ) x^{m}}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {c}\, \left (a^{2} x^{2}-1\right ) \left (1+m \right ) m}\) | \(101\) |
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Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=-\frac {{\left (a m x^{2} + {\left (m + 1\right )} x\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{m^{2} - {\left (a m^{2} + a m\right )} x + m} \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\text {Timed out} \]
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none
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.55 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\frac {{\left (a \sqrt {c} m x + \sqrt {c} {\left (m + 1\right )}\right )} {\left (a x + 1\right )} x^{m}}{{\left (m^{2} + m\right )} a^{2} x + {\left (m^{2} + m\right )} a} \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx=\int \frac {x^m\,\sqrt {c-\frac {c}{a^2\,x^2}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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