Optimal. Leaf size=13 \[ x-\frac {\tanh (a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8}
\begin {gather*} x-\frac {\tanh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rubi steps
\begin {align*} \int \tanh ^2(a+b x) \, dx &=-\frac {\tanh (a+b x)}{b}+\int 1 \, dx\\ &=x-\frac {\tanh (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.77 \begin {gather*} \frac {\tanh ^{-1}(\tanh (a+b x))}{b}-\frac {\tanh (a+b x)}{b} \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. \(35\) vs.
\(2(13)=26\).
time = 0.25, size = 36, normalized size = 2.77
method | result | size |
risch | \(x +\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )}\) | \(21\) |
derivativedivides | \(\frac {-\tanh \left (b x +a \right )-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}+\frac {\ln \left (\tanh \left (b x +a \right )+1\right )}{2}}{b}\) | \(36\) |
default | \(\frac {-\tanh \left (b x +a \right )-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}+\frac {\ln \left (\tanh \left (b x +a \right )+1\right )}{2}}{b}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 1.92 \begin {gather*} x + \frac {a}{b} - \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\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. 33 vs.
\(2 (13) = 26\).
time = 0.50, size = 33, normalized size = 2.54 \begin {gather*} \frac {{\left (b x + 1\right )} \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 15, normalized size = 1.15 \begin {gather*} \begin {cases} x - \frac {\tanh {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \tanh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 24, normalized size = 1.85 \begin {gather*} \frac {b x + a + \frac {2}{e^{\left (2 \, b x + 2 \, a\right )} + 1}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 13, normalized size = 1.00 \begin {gather*} x-\frac {\mathrm {tanh}\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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