Integrand size = 23, antiderivative size = 172 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}+\frac {5 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6311, 6316, 104, 163, 56, 221, 95, 212} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {5 \sqrt {a} \sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {a} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{x \sqrt {1-\frac {1}{a x}}} \]
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Rule 56
Rule 95
Rule 104
Rule 163
Rule 212
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^{3/2}} \, 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 {\left (1+\frac {x}{a}\right )^{3/2}}{\sqrt {x} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}+\frac {\left (a \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {-\frac {3}{2 a}-\frac {5 x}{2 a^2}}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}+\frac {\left (5 \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 )}{2 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}+\frac {\left (5 \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 )}{\sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}+\frac {5 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}+5 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )-4 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.47 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-4 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +5 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +\sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\, x}\) | \(117\) |
risch | \(-\frac {c \left (a x -1\right )}{x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {\left (\frac {4 a \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {5 a \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) c \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\) | \(140\) |
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Time = 0.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.27 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {4 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 5 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{2} - x\right )}}, -\frac {4 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 5 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {\sqrt {-a c x + c}}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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