Integrand size = 25, antiderivative size = 76 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6332, 6328, 45} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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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^2 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int x (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+a x^2\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.59 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2 (3+2 a x)}{6 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(\frac {x^{3} \left (2 a x +3\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{6 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(53\) |
default | \(\frac {x^{3} \left (2 a x +3\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{6 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(53\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.32 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {{\left (2 \, a x^{3} + 3 \, x^{2}\right )} \sqrt {a^{2} c}}{6 \, a^{2}} \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\text {Timed out} \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Time = 4.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.61 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (2\,a\,x+3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,\left (a\,x-1\right )} \]
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