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 = {3473, 8} \[ x-\frac {\tanh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
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 \[ \frac {\tanh ^{-1}(\tanh (a+b x))}{b}-\frac {\tanh (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 33, normalized size = 2.54 \[ \frac {{\left (b x + 1\right )} \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 24, normalized size = 1.85 \[ \frac {b x + a + \frac {2}{e^{\left (2 \, b x + 2 \, a\right )} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 18, normalized size = 1.38 \[ \frac {b x +a -\tanh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 1.92 \[ x + \frac {a}{b} - \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 20, normalized size = 1.54 \[ x+\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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