Integrand size = 24, antiderivative size = 152 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}}}+\frac {5 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.10 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6317, 6314, 100, 162, 65, 214, 212} \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {5 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right ) \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a x}}}+\frac {x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}} \]
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Rule 65
Rule 100
Rule 162
Rule 212
Rule 214
Rule 6314
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}} \, 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 )^{3/2}}{x^2 \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-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {5}{2 a}-\frac {3 x}{2 a^2}}{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-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}-\frac {\left (5 \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 )}{2 a \sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}}}-\frac {\left (5 \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 )}{\sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}} \\ & = \frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a x}}}+\frac {5 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a \sqrt {1-\frac {1}{a x}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.62 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} x+\frac {5 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a}-\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Time = 0.24 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-4 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}}\) | \(160\) |
risch | \(\frac {x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{a \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(205\) |
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Time = 0.32 (sec) , antiderivative size = 512, normalized size of antiderivative = 3.37 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {4 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 5 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {4 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 5 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {Timed out} \]
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\[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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