Integrand size = 27, antiderivative size = 303 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=-\frac {1115 \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {1115 \sqrt {c-\frac {c}{a x}} x}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6317, 6315, 91, 79, 44, 53, 65, 214} \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {1115 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {1115 \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {1115 x \sqrt {c-\frac {c}{a x}}}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {223 x^2 \sqrt {c-\frac {c}{a x}}}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {x^4 \sqrt {c-\frac {c}{a x}}}{4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {25 x^3 \sqrt {c-\frac {c}{a x}}}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 91
Rule 214
Rule 6315
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^3 \, dx}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^5 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {25}{2 a}+\frac {4 x}{a^2}}{x^4 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{4 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (223 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x^3 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{48 a^2 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {223 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{192 a^3 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {1115 \sqrt {c-\frac {c}{a x}} x}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{128 a^4 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {1115 \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {1115 \sqrt {c-\frac {c}{a x}} x}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a^4 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {1115 \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {1115 \sqrt {c-\frac {c}{a x}} x}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{64 a^3 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {1115 \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {1115 \sqrt {c-\frac {c}{a x}} x}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 1.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.55 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (-3345-1115 a x+446 a^2 x^2-200 a^3 x^3+48 a^4 x^4\right )}{-1+a^2 x^2}-3345 \sqrt {c} \log (1-a x)+3345 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{384 a^4} \]
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Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (96 a^{\frac {9}{2}} \sqrt {\left (a x +1\right ) x}\, x^{4}-400 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}+892 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-2230 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+3345 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x -6690 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+3345 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{384 \left (a x -1\right )^{2} a^{\frac {7}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(197\) |
risch | \(\frac {\left (48 a^{3} x^{3}-248 a^{2} x^{2}+694 a x -1809\right ) \left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{192 a^{3} \left (a x -1\right )}+\frac {\left (\frac {1115 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{128 a^{3} \sqrt {a^{2} c}}-\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a^{5} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a x -1}\) | \(209\) |
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Time = 0.29 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.17 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\left [\frac {3345 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{768 \, {\left (a^{5} x - a^{4}\right )}}, -\frac {3345 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} x - a^{4}\right )}}\right ] \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\text {Timed out} \]
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\[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int { \sqrt {c - \frac {c}{a x}} x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int x^3\,\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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