Integrand size = 14, antiderivative size = 47 \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6310, 6313, 272, 43, 65, 214} \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}-\frac {1}{2} a c x^2 \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\left ((a c) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right ) \\ & = (a c) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} (a c) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a} \\ & = -\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{2} (a c) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = -\frac {1}{2} a c \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {c \left (-a^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{2 a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(39)=78\).
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.96
method | result | size |
risch | \(-\frac {x \left (a x -1\right ) c}{2 \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(92\) |
default | \(-\frac {\left (a x -1\right ) c \left (x \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(93\) |
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.64 \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c x^{2} + a c x\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a} \]
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\[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=- c \left (\int \frac {a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (39) = 78\).
Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.81 \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {1}{2} \, a {\left (\frac {2 \, {\left (c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.23 \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {\sqrt {a^{2} x^{2} - 1} c x}{2 \, \mathrm {sgn}\left (a x + 1\right )} - \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{2 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 4.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.00 \[ \int e^{\coth ^{-1}(a x)} (c-a c x) \, dx=\frac {c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {c\,\sqrt {\frac {a\,x-1}{a\,x+1}}+c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}} \]
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