Optimal. Leaf size=16 \[ \tanh (x) \sqrt {-\coth ^2(x)} \log (\sinh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 16, 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 = {4121, 3658, 3475} \[ \tanh (x) \sqrt {-\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \sqrt {-1-\text {csch}^2(x)} \, dx &=\int \sqrt {-\coth ^2(x)} \, dx\\ &=\left (\sqrt {-\coth ^2(x)} \tanh (x)\right ) \int \coth (x) \, dx\\ &=\sqrt {-\coth ^2(x)} \log (\sinh (x)) \tanh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \tanh (x) \sqrt {-\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.45, size = 13, normalized size = 0.81 \[ -i \, x + i \, \log \left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.14, size = 32, normalized size = 2.00 \[ i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - i \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.31, size = 81, normalized size = 5.06 \[ -\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, x}{1+{\mathrm e}^{2 x}}+\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.41, size = 22, normalized size = 1.38 \[ -i \, x - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {-\frac {1}{{\mathrm {sinh}\relax (x)}^2}-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \operatorname {csch}^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________