Optimal. Leaf size=4 \[ x+\tanh (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {212}
\begin {gather*} x+\tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rubi steps
\begin {align*} \int \text {sech}^2(x) \left (1+\frac {1}{1-\tanh ^2(x)}\right ) \, dx &=\text {Subst}\left (\int \left (1+\frac {1}{1-x^2}\right ) \, dx,x,\tanh (x)\right )\\ &=\tanh (x)+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x+\tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} x+\tanh (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. \(33\) vs.
\(2(4)=8\).
time = 0.88, size = 34, normalized size = 8.50
method | result | size |
risch | \(x -\frac {2}{1+{\mathrm e}^{2 x}}\) | \(13\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {2 \tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}\) | \(34\) |
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 = 0.36, size = 14, normalized size = 3.50 \begin {gather*} \frac {{\left (x - 1\right )} \cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\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. 29 vs.
\(2 (3) = 6\).
time = 0.30, size = 29, normalized size = 7.25 \begin {gather*} - \frac {x \operatorname {sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} - 1} - \frac {\tanh {\left (x \right )} \operatorname {sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} - 1} \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 = 0.42, 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 = 1.72, 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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