Optimal. Leaf size=2 \[ \coth (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 2, 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 = {3254, 3852, 8}
\begin {gather*} \coth (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{1-\cosh ^2(x)} \, dx &=-\int \text {csch}^2(x) \, dx\\ &=i \text {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=\coth (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \coth (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. \(15\) vs.
\(2(2)=4\).
time = 0.45, size = 16, normalized size = 8.00
method | result | size |
risch | \(\frac {2}{{\mathrm e}^{2 x}-1}\) | \(11\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 10 vs.
\(2 (2) = 4\).
time = 0.26, size = 10, normalized size = 5.00 \begin {gather*} -\frac {2}{e^{\left (-2 \, x\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs.
\(2 (2) = 4\).
time = 0.36, size = 20, normalized size = 10.00 \begin {gather*} \frac {2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs.
\(2 (2) = 4\).
time = 0.21, size = 14, normalized size = 7.00 \begin {gather*} \frac {\tanh {\left (\frac {x}{2} \right )}}{2} + \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10 vs.
\(2 (2) = 4\).
time = 0.40, size = 10, normalized size = 5.00 \begin {gather*} \frac {2}{e^{\left (2 \, x\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 10, normalized size = 5.00 \begin {gather*} \frac {2}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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