Optimal. Leaf size=23 \[ -2 \sqrt {3} \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-1+i \text {csch}(x)}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3774, 207} \[ -2 \sqrt {3} \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-1+i \text {csch}(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 3774
Rubi steps
\begin {align*} \int \sqrt {-3+3 i \text {csch}(x)} \, dx &=-\left (6 i \operatorname {Subst}\left (\int \frac {1}{-3+x^2} \, dx,x,\frac {3 i \coth (x)}{\sqrt {-3+3 i \text {csch}(x)}}\right )\right )\\ &=-2 \sqrt {3} \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-1+i \text {csch}(x)}}\right )\\ \end {align*}
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Mathematica [A] time = 0.69, size = 46, normalized size = 2.00 \[ -\frac {2 \sqrt {3} \coth (x) \tanh ^{-1}\left (\sqrt {1+i \text {csch}(x)}\right )}{\sqrt {-1+i \text {csch}(x)} \sqrt {1+i \text {csch}(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 215, normalized size = 9.35 \[ \frac {1}{2} i \, \sqrt {3} \log \left ({\left (\frac {\sqrt {3} {\left (2 i \, \sqrt {3} e^{\left (2 \, x\right )} - 2 i \, \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 6 i \, e^{x} + 6\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} i \, \sqrt {3} \log \left ({\left (\frac {\sqrt {3} {\left (-2 i \, \sqrt {3} e^{\left (2 \, x\right )} + 2 i \, \sqrt {3}\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 6 i \, e^{x} + 6\right )} e^{\left (-x\right )}\right ) + \frac {1}{2} i \, \sqrt {3} \log \left ({\left (6 i \, \sqrt {3} e^{\left (2 \, x\right )} - 6 \, \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (6 i \, e^{\left (3 \, x\right )} - 12 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} + 12\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} - 12 i \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{2} i \, \sqrt {3} \log \left ({\left (-6 i \, \sqrt {3} e^{\left (2 \, x\right )} + 6 \, \sqrt {3} e^{x} + \frac {\sqrt {3} {\left (6 i \, e^{\left (3 \, x\right )} - 12 \, e^{\left (2 \, x\right )} - 6 i \, e^{x} + 12\right )}}{\sqrt {e^{\left (2 \, x\right )} - 1}} + 12 i \, \sqrt {3}\right )} e^{\left (-2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 i \, \operatorname {csch}\relax (x) - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \sqrt {-3+3 i \mathrm {csch}\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 i \, \operatorname {csch}\relax (x) - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {-3+\frac {3{}\mathrm {i}}{\mathrm {sinh}\relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt {3} \int \sqrt {i \operatorname {csch}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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