Optimal. Leaf size=13 \[ \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4122, 191} \[ \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx &=\operatorname {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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.05, size = 14, normalized size = 1.08 \[ \frac {1}{2} \, {\left (-i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.13, size = 25, normalized size = 1.92 \[ -\frac {-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 58, normalized size = 4.46 \[ \frac {{\mathrm e}^{2 x}}{2 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {1}{2 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.49, size = 11, normalized size = 0.85 \[ \frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{2} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 31, normalized size = 2.38 \[ -{\mathrm {e}}^{-2\,x}\,\sqrt {-\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}}\,\left (\frac {{\mathrm {e}}^{4\,x}}{4}-\frac {1}{4}\right ) \]
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 \[ \int \frac {1}{\sqrt {- \operatorname {csch}^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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