Optimal. Leaf size=2 \[ \tanh (x) \]
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Rubi [A]
time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3852, 8}
\begin {gather*} \tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=i \text {Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=\tanh (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 2, normalized size = 1.00 \begin {gather*} \tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 3, normalized size = 1.50
method | result | size |
default | \(\tanh \left (x \right )\) | \(3\) |
parallelrisch | \(\tanh \left (x \right )\) | \(3\) |
risch | \(-\frac {2}{1+{\mathrm e}^{2 x}}\) | \(11\) |
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 = 0.34, 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 = 0.55, 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 = 0.32, size = 14, normalized size = 7.00 \begin {gather*} \frac {2 \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \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. 10 vs.
\(2 (2) = 4\).
time = 0.46, 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 = 0.01, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {tanh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 9, normalized size = 4.50 \begin {gather*} \tanh \left (x \right )-\frac {\left (\tanh ^{3}\left (x \right )\right )}{3} \end {gather*}
Warning: Unable to verify antiderivative.
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