Integrand size = 10, antiderivative size = 13 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4207, 197} \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Rule 197
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(11)=22\).
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 4.46
method | result | size |
risch | \(\frac {{\mathrm e}^{2 x}}{2 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {1}{2 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\) | \(58\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {1}{2} \, {\left (-i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \]
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\[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\int \frac {1}{\sqrt {- \operatorname {csch}^{2}{\left (x \right )}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{2} i \, e^{x} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=-\frac {-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]
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Time = 2.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=-{\mathrm {e}}^{-2\,x}\,\sqrt {-\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}}\,\left (\frac {{\mathrm {e}}^{4\,x}}{4}-\frac {1}{4}\right ) \]
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