Optimal. Leaf size=13 \[ \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3657, 4122, 191} \[ \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 1, normalized size = 0.08 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.12, size = 24, normalized size = 1.85 \[ -\frac {-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 14, normalized size = 1.08 \[ \frac {\coth \relax (x )}{\sqrt {1-\left (\coth ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.42, 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.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.34, size = 18, normalized size = 1.38 \[ -\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)\,\sqrt {-\frac {1}{{\mathrm {cosh}\relax (x)}^2-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - \coth ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________