Integrand size = 6, antiderivative size = 24 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=e^{\text {csch}^{-1}(a x)} x-\frac {\text {csch}^{-1}(a x)}{a}+\frac {\log (x)}{a} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6466, 29, 248, 283, 221} \[ \int e^{\text {csch}^{-1}(a x)} \, dx=x \sqrt {\frac {1}{a^2 x^2}+1}+\frac {\log (x)}{a}-\frac {\text {csch}^{-1}(a x)}{a} \]
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Rule 29
Rule 221
Rule 248
Rule 283
Rule 6466
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{x} \, dx}{a}+\int \sqrt {1+\frac {1}{a^2 x^2}} \, dx \\ & = \frac {\log (x)}{a}-\text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \sqrt {1+\frac {1}{a^2 x^2}} x+\frac {\log (x)}{a}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2} \\ & = \sqrt {1+\frac {1}{a^2 x^2}} x-\frac {\text {csch}^{-1}(a x)}{a}+\frac {\log (x)}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {a \sqrt {1+\frac {1}{a^2 x^2}} x-\text {arcsinh}\left (\frac {1}{a x}\right )+\log (a x)}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(37)=74\).
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71
method | result | size |
default | \(-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right )\right )}{\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}}+\frac {\ln \left (x \right )}{a}\) | \(113\) |
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (37) = 74\).
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + \log \left (x\right )}{a} \]
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Time = 0.76 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {x}{\sqrt {1 + \frac {1}{a^{2} x^{2}}}} + \frac {\log {\left (x \right )}}{a} - \frac {\operatorname {asinh}{\left (\frac {1}{a x} \right )}}{a} + \frac {1}{a^{2} x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]
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none
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.67 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{2 \, a} + \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} + \frac {\log \left (x\right )}{a} \]
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none
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=-\frac {{\left (\log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} \mathrm {sgn}\left (x\right )\right )} {\left | a \right |}}{2 \, a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a} \]
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Time = 4.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int e^{\text {csch}^{-1}(a x)} \, dx=\frac {\ln \left (x\right )}{a}+x\,\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {\mathrm {asin}\left (\frac {1{}\mathrm {i}}{a\,x}\right )\,1{}\mathrm {i}}{a} \]
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