Integrand size = 27, antiderivative size = 124 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \arcsin (a x)}{a^2 \sqrt {1-a x} \sqrt {1+a x}} \]
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Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6294, 6264, 81, 52, 41, 222} \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {x \arcsin (a x) \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {a x+1} \sqrt {1-a x}}+\frac {x (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^2}+\frac {x \sqrt {c-\frac {c}{a^2 x^2}}}{a^2} \]
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Rule 41
Rule 52
Rule 81
Rule 222
Rule 6264
Rule 6294
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx \\ & = -\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-2 \text {arctanh}(a x)} x \sqrt {1-a x} \sqrt {1+a x} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {x (1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{\sqrt {1+a x}} \, dx}{3 a \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)^2}{3 a^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \arcsin (a x)}{a^2 \sqrt {1-a x} \sqrt {1+a x}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2} \left (5-3 a x+a^2 x^2\right )-3 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{3 a^2 \sqrt {-1+a^2 x^2}} \]
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Time = 0.57 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {\left (a^{2} x^{2}-3 a x +5\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{3 a^{2}}-\frac {\ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{a \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\) | \(125\) |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left ({\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3}-3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x +3 c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-6 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right )+6 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c \right )}{3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c}\) | \(173\) |
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Time = 0.26 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.65 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\left [\frac {2 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{6 \, a^{3}}, \frac {{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 3 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{3 \, a^{3}}\right ] \]
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\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
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\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{a x + 1} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\frac {1}{6} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (x {\left (\frac {x \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {3 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} + \frac {5 \, \mathrm {sgn}\left (x\right )}{a^{4}}\right )} + \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{3} {\left | a \right |}} - \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 10 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{4} {\left | a \right |}}\right )} {\left | a \right |} \]
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Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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