Integrand size = 24, antiderivative size = 118 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}} \]
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Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6312, 893, 883, 889, 214} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=-\frac {5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}}+\frac {x \sqrt {c-\frac {c}{a x}}}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 214
Rule 883
Rule 889
Rule 893
Rule 6312
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3} \\ & = \frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a c^2} \\ & = \frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \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 )}{2 a c} \\ & = \frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {(5 c) \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 )}{a^3} \\ & = \frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (a x+5 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {1}{a x}\right )\right )}{a \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}} \]
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Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x -10 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, c \sqrt {\left (a x +1\right ) x}}\) | \(149\) |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (-\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a \sqrt {a^{2} c}}+\frac {4 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a^{3} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(174\) |
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Time = 0.30 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.57 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [\frac {5 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c x - a c\right )}}, \frac {5 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c x - a c\right )}}\right ] \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {c - \frac {c}{a x}}} \,d x } \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {c - \frac {c}{a x}}} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\sqrt {c-\frac {c}{a\,x}}} \,d x \]
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