Integrand size = 12, antiderivative size = 23 \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-2 \sqrt {3} \arctan \left (\frac {\coth (x)}{\sqrt {-1-i \text {csch}(x)}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3859, 213} \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-2 \sqrt {3} \arctan \left (\frac {\coth (x)}{\sqrt {-1-i \text {csch}(x)}}\right ) \]
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Rule 213
Rule 3859
Rubi steps \begin{align*} \text {integral}& = 6 i \text {Subst}\left (\int \frac {1}{-3+x^2} \, dx,x,-\frac {3 i \coth (x)}{\sqrt {-3-3 i \text {csch}(x)}}\right ) \\ & = -2 \sqrt {3} \arctan \left (\frac {\coth (x)}{\sqrt {-1-i \text {csch}(x)}}\right ) \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-\frac {2 \sqrt {3} \text {arctanh}\left (\sqrt {1-i \text {csch}(x)}\right ) \coth (x)}{\sqrt {-1-i \text {csch}(x)} \sqrt {1-i \text {csch}(x)}} \]
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\[\int \sqrt {-3-3 i \operatorname {csch}\left (x \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 9.48 \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=-\frac {1}{2} i \, \sqrt {3} \log \left (-2 \, {\left (\frac {\sqrt {3} {\left (i \, \sqrt {3} e^{\left (2 \, x\right )} - i \, \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 3 i \, e^{x} + 3\right )} e^{\left (-x\right )}\right ) + \frac {1}{2} i \, \sqrt {3} \log \left (-2 \, {\left (\frac {\sqrt {3} {\left (-i \, \sqrt {3} e^{\left (2 \, x\right )} + i \, \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 3 i \, e^{x} + 3\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} i \, \sqrt {3} \log \left (-6 \, {\left (i \, \sqrt {3} e^{\left (2 \, x\right )} + \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (-i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 2 i \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{2} i \, \sqrt {3} \log \left (-6 \, {\left (-i \, \sqrt {3} e^{\left (2 \, x\right )} - \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (-i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 2 i \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) \]
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\[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\sqrt {3} \int \sqrt {- i \operatorname {csch}{\left (x \right )} - 1}\, dx \]
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\[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\int { \sqrt {-3 i \, \operatorname {csch}\left (x\right ) - 3} \,d x } \]
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\[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\int { \sqrt {-3 i \, \operatorname {csch}\left (x\right ) - 3} \,d x } \]
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Timed out. \[ \int \sqrt {-3-3 i \text {csch}(x)} \, dx=\int \sqrt {-3-\frac {3{}\mathrm {i}}{\mathrm {sinh}\left (x\right )}} \,d x \]
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