Optimal. Leaf size=11 \[ \coth (x)-\frac {\coth ^2(x)}{2} \]
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Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3568}
\begin {gather*} \coth (x)-\frac {\coth ^2(x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3568
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(x)}{1+\coth (x)} \, dx &=\text {Subst}(\int (1-x) \, dx,x,\coth (x))\\ &=\coth (x)-\frac {\coth ^2(x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} \coth (x)-\frac {\text {csch}^2(x)}{2} \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 = 0.61, size = 32, normalized size = 2.91
method | result | size |
risch | \(-\frac {2}{\left ({\mathrm e}^{2 x}-1\right )^{2}}\) | \(11\) |
default | \(-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}\) | \(32\) |
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. 41 vs.
\(2 (9) = 18\).
time = 0.26, size = 41, normalized size = 3.73 \begin {gather*} \frac {4 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, 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. 55 vs.
\(2 (9) = 18\).
time = 0.35, size = 55, normalized size = 5.00 \begin {gather*} -\frac {2}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \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 \frac {\operatorname {csch}^{4}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 10, normalized size = 0.91 \begin {gather*} -\frac {2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 16, normalized size = 1.45 \begin {gather*} -\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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