Integrand size = 27, antiderivative size = 76 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.17 (sec) , antiderivative size = 76, 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} \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {x^3 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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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^3 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int x^2 (-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^2+a x^3\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {x^2 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.59 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {x^2 (-4+3 a x) \sqrt {c-a^2 c x^2}}{12 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.54 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(\frac {x^{3} \left (3 a x -4\right ) \sqrt {-a^{2} c \,x^{2}+c}\, \sqrt {\frac {a x -1}{a x +1}}}{12 a x -12}\) | \(47\) |
default | \(\frac {\left (3 a x -4\right ) x^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{12 a x -12}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.33 \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\frac {{\left (3 \, a x^{4} - 4 \, x^{3}\right )} \sqrt {-a^{2} c}}{12 \, a} \]
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\[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int x^{2} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}\, dx \]
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\[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{2} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} x^{2} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a^2 c x^2} \, dx=\int x^2\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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