Optimal. Leaf size=11 \[ \coth (x)-\frac {\coth ^3(x)}{3} \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3254, 3852}
\begin {gather*} \coth (x)-\frac {\coth ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{\left (1-\cosh ^2(x)\right )^2} \, dx &=\int \text {csch}^4(x) \, dx\\ &=i \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )\\ &=\coth (x)-\frac {\coth ^3(x)}{3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.55 \begin {gather*} \frac {2 \coth (x)}{3}-\frac {1}{3} \coth (x) \text {csch}^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. \(31\) vs.
\(2(9)=18\).
time = 0.45, size = 32, normalized size = 2.91
method | result | size |
risch | \(-\frac {4 \left (3 \,{\mathrm e}^{2 x}-1\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}\) | \(19\) |
default | \(-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {3 \tanh \left (\frac {x}{2}\right )}{8}-\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {3}{8 \tanh \left (\frac {x}{2}\right )}\) | \(32\) |
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. 49 vs.
\(2 (9) = 18\).
time = 0.27, size = 49, normalized size = 4.45 \begin {gather*} \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac {4}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \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. 84 vs.
\(2 (9) = 18\).
time = 0.39, size = 84, normalized size = 7.64 \begin {gather*} -\frac {8 \, {\left (\cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right )}}{3 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + {\left (10 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{3} + {\left (10 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right )\right )}} \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. 34 vs.
\(2 (8) = 16\).
time = 0.52, size = 34, normalized size = 3.09 \begin {gather*} - \frac {\tanh ^{3}{\left (\frac {x}{2} \right )}}{24} + \frac {3 \tanh {\left (\frac {x}{2} \right )}}{8} + \frac {3}{8 \tanh {\left (\frac {x}{2} \right )}} - \frac {1}{24 \tanh ^{3}{\left (\frac {x}{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 1.64 \begin {gather*} -\frac {4 \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 18, normalized size = 1.64 \begin {gather*} -\frac {4\,\left (3\,{\mathrm {e}}^{2\,x}-1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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