Optimal. Leaf size=13 \[ x-\frac {\coth (a+b x)}{b} \]
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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} \[ x-\frac {\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \coth ^2(a+b x) \, dx &=-\frac {\coth (a+b x)}{b}+\int 1 \, dx\\ &=x-\frac {\coth (a+b x)}{b}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 2.08 \[ -\frac {\coth (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 33, normalized size = 2.54 \[ \frac {{\left (b x + 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, 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 [B] time = 0.01, size = 41, normalized size = 3.15 \[ -\frac {\coth \left (b x +a \right )}{b}-\frac {\ln \left (\coth \left (b x +a \right )-1\right )}{2 b}+\frac {\ln \left (\coth \left (b x +a \right )+1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, 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 = 1.02, size = 13, normalized size = 1.00 \[ x-\frac {\mathrm {coth}\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.26, size = 36, normalized size = 2.77 \[ \begin {cases} x \coth ^{2}{\relax (a )} & \text {for}\: b = 0 \\\tilde {\infty } x & \text {for}\: a = \log {\left (- e^{- b x} \right )} \vee a = \log {\left (e^{- b x} \right )} \\x - \frac {1}{b \tanh {\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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