Integrand size = 23, antiderivative size = 238 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {119 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.19 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6311, 6316, 91, 81, 52, 56, 221} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {119 a^{5/2} \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}}}-\frac {119 a^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{8 x \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 x^3 \sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-a c x}}{x^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {119 a \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{12 x^2 \sqrt {1-\frac {1}{a x}}} \]
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Rule 52
Rule 56
Rule 81
Rule 91
Rule 221
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^{7/2}} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{3/2} \left (1-\frac {x}{a}\right )^2}{\left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}+\frac {\left (2 a^2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{3/2} \left (\frac {19}{2 a^2}-\frac {x}{2 a^3}\right )}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {\left (119 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {\left (119 a \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (119 a^2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {\left (119 a^2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {\frac {1}{x}}\right )}{8 \sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {119 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c-a c x} \left (-8+38 a x-119 a^2 x^2-357 a^3 x^3+\frac {357 a^{7/2} \sqrt {1+\frac {1}{a x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{7/2}}\right )}{24 a \sqrt {1-\frac {1}{a^2 x^2}} x^4} \]
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Time = 0.47 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.48
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (357 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3} \sqrt {-c \left (a x +1\right )}+357 a^{3} x^{3} \sqrt {c}+119 \sqrt {c}\, a^{2} x^{2}-38 \sqrt {c}\, a x +8 \sqrt {c}\right )}{24 \left (a x -1\right )^{2} \sqrt {c}\, x^{3}}\) | \(114\) |
risch | \(\frac {\left (165 a^{3} x^{3}+119 a^{2} x^{2}-38 a x +8\right ) c \sqrt {\frac {a x -1}{a x +1}}}{24 x^{3} \sqrt {-c \left (a x -1\right )}}-\frac {\left (-\frac {8 a^{3}}{\sqrt {-a c x -c}}-\frac {119 a^{3} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(134\) |
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Time = 0.26 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^4} \,d x \]
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