Integrand size = 27, antiderivative size = 147 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {45 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]
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Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6302, 6268, 25, 528, 457, 100, 156, 162, 65, 214} \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {45 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3}+\frac {19 x \sqrt {c-\frac {c}{a x}}}{8 a^2}+\frac {1}{3} x^3 \sqrt {c-\frac {c}{a x}}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{12 a} \]
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Rule 25
Rule 65
Rule 100
Rule 156
Rule 162
Rule 214
Rule 457
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} x^2 (1-a x)}{1+a x} \, dx \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^3}{1+a x} \, dx}{c} \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{a+\frac {1}{x}} \, dx}{c} \\ & = -\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^4 (a+x)} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {\text {Subst}\left (\int \frac {\frac {13 c^2}{2}-\frac {11 c^2 x}{2 a}}{x^3 (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c} \\ & = -\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {\text {Subst}\left (\int \frac {\frac {57 c^3}{4}-\frac {39 c^3 x}{4 a}}{x^2 (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 a c^2} \\ & = \frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {\text {Subst}\left (\int \frac {\frac {135 c^4}{8}-\frac {57 c^4 x}{8 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 a^2 c^3} \\ & = \frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {(45 c) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^3}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^3} \\ & = \frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {45 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{8 a^2}+\frac {8 \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{a^2} \\ & = \frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {45 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {a \sqrt {c-\frac {c}{a x}} x \left (57-26 a x+8 a^2 x^2\right )-135 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )+96 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{24 a^3} \]
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Time = 0.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {\left (8 a^{2} x^{2}-26 a x +57\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2}}+\frac {\left (-\frac {45 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{16 a^{2} \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{a^{3} \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{a x -1}\) | \(189\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}-36 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x +96 \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}+18 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}-96 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right )-144 a^{2} \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2}\right )}{48 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}}\) | \(237\) |
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Time = 0.26 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.76 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {96 \, \sqrt {2} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 2 \, {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 135 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{48 \, a^{3}}, -\frac {96 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 135 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\right ] \]
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\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
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\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}} x^{2}}{a x + 1} \,d x } \]
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Exception generated. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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