Integrand size = 24, antiderivative size = 50 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\sqrt {c-\frac {c}{a x}} x+\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
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Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6268, 25, 528, 382, 79, 65, 214} \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+x \sqrt {c-\frac {c}{a x}} \]
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Rule 25
Rule 65
Rule 79
Rule 214
Rule 382
Rule 528
Rule 6268
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx \\ & = -\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{1-a x} \, dx \\ & = \frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x} \, dx}{a} \\ & = \frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}}} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {a+x}{x^2 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \sqrt {c-\frac {c}{a x}} x-\frac {(3 c) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \sqrt {c-\frac {c}{a x}} x+3 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right ) \\ & = \sqrt {c-\frac {c}{a x}} x+\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\sqrt {c-\frac {c}{a x}} x+\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(42)=84\).
Time = 0.49 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.96
method | result | size |
risch | \(x \sqrt {\frac {c \left (a x -1\right )}{a x}}+\frac {3 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(98\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {a \,x^{2}-x}\, \sqrt {a}-4 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-2 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}}\) | \(120\) |
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none
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.48 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} + 3 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a}, \frac {a x \sqrt {\frac {a c x - c}{a x}} - 3 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{a}\right ] \]
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\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
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\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{a x - 1} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (42) = 84\).
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.92 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {3 \, \sqrt {c} \log \left ({\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, a} - \frac {3 \, \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} \sqrt {c} {\left | a \right |} + a c \right |}\right )}{2 \, a \mathrm {sgn}\left (x\right )} + \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |}}{a^{2} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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