Optimal. Leaf size=11 \[ \frac {\coth (x)}{\sqrt {\text {csch}^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3657, 4122, 191} \[ \frac {\coth (x)}{\sqrt {\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\coth ^2(x)}} \, dx &=\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 = 11, normalized size = 1.00 \[ \frac {\coth (x)}{\sqrt {\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 2, normalized size = 0.18 \[ \cosh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 18, normalized size = 1.64 \[ \frac {e^{\left (-x\right )} + e^{x}}{2 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 12, normalized size = 1.09 \[ \frac {\coth \relax (x )}{\sqrt {-1+\coth ^{2}\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 11, normalized size = 1.00 \[ -\frac {1}{2} \, e^{\left (-x\right )} - \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 15, normalized size = 1.36 \[ \mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)\,\sqrt {\frac {1}{{\mathrm {cosh}\relax (x)}^2-1}} \]
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 \[ \int \frac {1}{\sqrt {\coth ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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