Optimal. Leaf size=39 \[ \frac {a x}{a^2-b^2}-\frac {b \log (b \cosh (x)+a \sinh (x))}{a^2-b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 39, 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 = {3565, 3611}
\begin {gather*} \frac {a x}{a^2-b^2}-\frac {b \log (a \sinh (x)+b \cosh (x))}{a^2-b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3565
Rule 3611
Rubi steps
\begin {align*} \int \frac {1}{a+b \coth (x)} \, dx &=\frac {a x}{a^2-b^2}-\frac {(i b) \int \frac {-i b-i a \coth (x)}{a+b \coth (x)} \, dx}{a^2-b^2}\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (b \cosh (x)+a \sinh (x))}{a^2-b^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 29, normalized size = 0.74 \begin {gather*} \frac {a x-b \log (b \cosh (x)+a \sinh (x))}{a^2-b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 55, normalized size = 1.41
method | result | size |
derivativedivides | \(-\frac {b \ln \left (a +b \coth \left (x \right )\right )}{\left (a -b \right ) \left (a +b \right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2 b +2 a}\) | \(55\) |
default | \(-\frac {b \ln \left (a +b \coth \left (x \right )\right )}{\left (a -b \right ) \left (a +b \right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2 b +2 a}\) | \(55\) |
risch | \(\frac {x}{a +b}+\frac {2 b x}{a^{2}-b^{2}}-\frac {b \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{2}-b^{2}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 37, normalized size = 0.95 \begin {gather*} -\frac {b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{2} - b^{2}} + \frac {x}{a + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 42, normalized size = 1.08 \begin {gather*} \frac {{\left (a + b\right )} x - b \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} - b^{2}} \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. 148 vs.
\(2 (29) = 58\).
time = 0.50, size = 148, normalized size = 3.79 \begin {gather*} \begin {cases} \tilde {\infty } \left (x - \log {\left (\tanh {\left (x \right )} + 1 \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {x - \log {\left (\tanh {\left (x \right )} + 1 \right )}}{b} & \text {for}\: a = 0 \\- \frac {x \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} - 2 b} + \frac {x}{2 b \tanh {\left (x \right )} - 2 b} - \frac {1}{2 b \tanh {\left (x \right )} - 2 b} & \text {for}\: a = - b \\\frac {x \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} + 2 b} + \frac {x}{2 b \tanh {\left (x \right )} + 2 b} + \frac {1}{2 b \tanh {\left (x \right )} + 2 b} & \text {for}\: a = b \\\frac {a x}{a^{2} - b^{2}} - \frac {b x}{a^{2} - b^{2}} + \frac {b \log {\left (\tanh {\left (x \right )} + 1 \right )}}{a^{2} - b^{2}} - \frac {b \log {\left (\tanh {\left (x \right )} + \frac {b}{a} \right )}}{a^{2} - b^{2}} & \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 = 43, normalized size = 1.10 \begin {gather*} -\frac {b \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{2} - b^{2}} + \frac {x}{a - b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 42, normalized size = 1.08 \begin {gather*} \frac {x}{a-b}-\frac {b\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^2-b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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