Optimal. Leaf size=4 \[ x+\coth (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {464, 212}
\begin {gather*} x+\coth (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 464
Rubi steps
\begin {align*} \int \left (-1-\frac {1}{1-\coth ^2(x)}\right ) \text {csch}^2(x) \, dx &=-\text {Subst}\left (\int \frac {1-2 x^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\coth (x)+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x+\coth (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} x+\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. \(31\) vs.
\(2(4)=8\).
time = 0.82, size = 32, normalized size = 8.00
method | result | size |
risch | \(x +\frac {2}{{\mathrm e}^{2 x}-1}\) | \(13\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {1}{2 \tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(32\) |
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. 12 vs.
\(2 (4) = 8\).
time = 0.26, size = 12, normalized size = 3.00 \begin {gather*} x - \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. 14 vs.
\(2 (4) = 8\).
time = 0.32, size = 14, normalized size = 3.50 \begin {gather*} \frac {{\left (x - 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}{\sinh \left (x\right )} \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 \left (- \frac {2 \operatorname {csch}^{2}{\left (x \right )}}{\coth ^{2}{\left (x \right )} - 1}\right )\, dx - \int \frac {\coth ^{2}{\left (x \right )} \operatorname {csch}^{2}{\left (x \right )}}{\coth ^{2}{\left (x \right )} - 1}\, dx \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. 12 vs.
\(2 (4) = 8\).
time = 0.40, size = 12, normalized size = 3.00 \begin {gather*} x + \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 = 1.83, size = 12, normalized size = 3.00 \begin {gather*} x+\frac {2}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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