Optimal. Leaf size=16 \[ \coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, 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 = {4206, 3739,
3556} \begin {gather*} \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3739
Rule 4206
Rubi steps
\begin {align*} \int \sqrt {-1+\text {sech}^2(x)} \, dx &=\int \sqrt {-\tanh ^2(x)} \, dx\\ &=\left (\coth (x) \sqrt {-\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs.
\(2(14)=28\).
time = 1.44, size = 81, normalized size = 5.06
method | result | size |
risch | \(-\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 13, normalized size = 0.81 \begin {gather*} -i \, x - i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {\operatorname {sech}^{2}{\left (x \right )} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.39, size = 31, normalized size = 1.94 \begin {gather*} i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \sqrt {\frac {1}{{\mathrm {cosh}\left (x\right )}^2}-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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