Integrand size = 18, antiderivative size = 80 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6310, 6313, 866, 1819, 12, 272, 65, 214} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c}+\frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 866
Rule 1819
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right ) x} \, dx}{a c} \\ & = \frac {\text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x \left (1-\frac {x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^4}{x \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3-\frac {4 x}{a}+\frac {3 x^2}{a^2}}{x \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a c} \\ & = \frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {\text {Subst}\left (\int \frac {3}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 a c} \\ & = \frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c} \\ & = \frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {a \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c} \\ & = \frac {8 \left (a+\frac {1}{x}\right )}{3 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\frac {4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+2 a x)}{(-1+a x)^2}-\frac {3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{3 c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(70)=140\).
Time = 0.39 (sec) , antiderivative size = 345, normalized size of antiderivative = 4.31
method | result | size |
default | \(-\frac {3 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-3 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+9 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -3 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{3 a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(345\) |
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Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.50 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - 4 \, {\left (2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {1}{\frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \]
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Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {1}{3} \, a {\left (\frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {2 \, {\left (\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\frac {2\,\left (a\,x-1\right )}{a\,x+1}+\frac {2}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \]
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