Integrand size = 18, antiderivative size = 33 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\text {csch}^{-1}(c x)}{c}+\frac {\log (x)}{c}-\frac {\log \left (1+c^2 x^2\right )}{2 c} \]
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Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6475, 342, 221, 272, 36, 29, 31} \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\log \left (c^2 x^2+1\right )}{2 c}+\frac {\log (x)}{c}-\frac {\text {csch}^{-1}(c x)}{c} \]
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Rule 29
Rule 31
Rule 36
Rule 221
Rule 272
Rule 342
Rule 6475
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c^2}+\frac {\int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{c} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c^2}+\frac {\text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 c} \\ & = -\frac {\text {csch}^{-1}(c x)}{c}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}-\frac {1}{2} c \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {\text {csch}^{-1}(c x)}{c}+\frac {\log (x)}{c}-\frac {\log \left (1+c^2 x^2\right )}{2 c} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\text {arcsinh}\left (\frac {1}{c x}\right )}{c}+\frac {\log (x)}{c}-\frac {\log \left (1+c^2 x^2\right )}{2 c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(31)=62\).
Time = 0.88 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.15
method | result | size |
default | \(\frac {\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \left (\sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}-\sqrt {-\frac {\left (-c^{2} x +\sqrt {-c^{2}}\right ) \left (c^{2} x +\sqrt {-c^{2}}\right )}{c^{4}}}\, c^{2} \sqrt {\frac {1}{c^{2}}}-\ln \left (\frac {2 \sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}+2}{c^{2} x}\right )\right )}{\sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}}+\frac {\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}}{c}\) | \(170\) |
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (31) = 62\).
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\log \left (c^{2} x^{2} + 1\right ) + 2 \, \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 2 \, \log \left (x\right )}{2 \, c} \]
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\[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=\frac {\int \frac {c x \sqrt {1 + \frac {1}{c^{2} x^{2}}}}{c^{2} x^{3} + x}\, dx + \int \frac {1}{c^{2} x^{3} + x}\, dx}{c} \]
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\[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=\int { \frac {\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}}{c^{2} x^{2} + 1} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (31) = 62\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\log \left (c^{2} x^{2} + 1\right )}{2 \, c} - \frac {{\left ({\left | c \right |} \mathrm {sgn}\left (x\right ) - c\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{2 \, c^{2}} + \frac {{\left ({\left | c \right |} \mathrm {sgn}\left (x\right ) + c\right )} \log \left (\sqrt {c^{2} x^{2} + 1} - 1\right )}{2 \, c^{2}} \]
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Time = 5.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{c^2}}}{x}\right )\,\sqrt {\frac {1}{c^2}}-\frac {\ln \left (c^2\,x^2+1\right )-2\,\ln \left (x\right )}{2\,c} \]
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