Optimal. Leaf size=39 \[ \frac {a x}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]
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Rubi [A] time = 0.05, 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 = {3484, 3530} \[ \frac {a x}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]
Antiderivative was successfully verified.
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Rule 3484
Rule 3530
Rubi steps
\begin {align*} \int \frac {1}{a+b \tanh (x)} \, dx &=\frac {a x}{a^2-b^2}-\frac {(i b) \int \frac {-i b-i a \tanh (x)}{a+b \tanh (x)} \, dx}{a^2-b^2}\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 29, normalized size = 0.74 \[ \frac {a x-b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a+b \tanh (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.33, size = 42, normalized size = 1.08 \[ \frac {{\left (a + b\right )} x - b \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 43, normalized size = 1.10 \[ -\frac {b \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} - b^{2}} + \frac {x}{a - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 55, normalized size = 1.41
method | result | size |
derivativedivides | \(\frac {\ln \left (1+\tanh \relax (x )\right )}{2 a -2 b}-\frac {b \ln \left (a +b \tanh \relax (x )\right )}{\left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\tanh \relax (x )-1\right )}{2 a +2 b}\) | \(55\) |
default | \(\frac {\ln \left (1+\tanh \relax (x )\right )}{2 a -2 b}-\frac {b \ln \left (a +b \tanh \relax (x )\right )}{\left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\tanh \relax (x )-1\right )}{2 a +2 b}\) | \(55\) |
risch | \(\frac {x}{a +b}+\frac {2 b x}{a^{2}-b^{2}}-\frac {b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{2}-b^{2}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 41, normalized size = 1.05 \[ -\frac {b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} - b^{2}} + \frac {x}{a + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 35, normalized size = 0.90 \[ \frac {a\,x-b\,\left (x-\ln \left (\mathrm {tanh}\relax (x)+1\right )+\ln \left (a+b\,\mathrm {tanh}\relax (x)\right )\right )}{a^2-b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 146, normalized size = 3.74 \[ \begin {cases} \tilde {\infty } \left (x - \log {\left (\tanh {\relax (x )} + 1 \right )} + \log {\left (\tanh {\relax (x )} \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{a} & \text {for}\: b = 0 \\- \frac {x \tanh {\relax (x )}}{2 b \tanh {\relax (x )} - 2 b} + \frac {x}{2 b \tanh {\relax (x )} - 2 b} + \frac {1}{2 b \tanh {\relax (x )} - 2 b} & \text {for}\: a = - b \\\frac {x \tanh {\relax (x )}}{2 b \tanh {\relax (x )} + 2 b} + \frac {x}{2 b \tanh {\relax (x )} + 2 b} - \frac {1}{2 b \tanh {\relax (x )} + 2 b} & \text {for}\: a = b \\\frac {a x}{a^{2} - b^{2}} - \frac {b x}{a^{2} - b^{2}} - \frac {b \log {\left (\frac {a}{b} + \tanh {\relax (x )} \right )}}{a^{2} - b^{2}} + \frac {b \log {\left (\tanh {\relax (x )} + 1 \right )}}{a^{2} - b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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