Integrand size = 25, antiderivative size = 164 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \]
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Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6313, 877, 887, 889, 214} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{12 a \sqrt {c-\frac {c}{a x}}}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \]
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Rule 214
Rule 877
Rule 887
Rule 889
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^4 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {\text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 a} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = -\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {\text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^3} \\ & = -\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {c^2 \text {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^5} \\ & = -\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.90 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (-3+2 a x+8 a^2 x^2\right )}{-1+a x}-3 \sqrt {c} \log (1-a x)+3 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{48 a^3} \]
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Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}+4 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-6 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{48 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {5}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(121\) |
risch | \(\frac {\left (8 a^{2} x^{2}+2 a x -3\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{16 a^{2} \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(148\) |
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Time = 0.29 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.05 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}} x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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