Optimal. Leaf size=64 \[ -\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}+\frac {a^3 \log (a+b \tanh (x))}{b^2 \left (a^2-b^2\right )}-\frac {\tanh (x)}{b} \]
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Rubi [A] time = 0.13, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3566, 3626, 3617, 31, 3475} \[ -\frac {b x}{a^2-b^2}+\frac {a^3 \log (a+b \tanh (x))}{b^2 \left (a^2-b^2\right )}+\frac {a \log (\cosh (x))}{a^2-b^2}-\frac {\tanh (x)}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3566
Rule 3617
Rule 3626
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{a+b \tanh (x)} \, dx &=-\frac {\tanh (x)}{b}-\frac {\int \frac {-a-b \tanh (x)+a \tanh ^2(x)}{a+b \tanh (x)} \, dx}{b}\\ &=-\frac {b x}{a^2-b^2}-\frac {\tanh (x)}{b}+\frac {a \int \tanh (x) \, dx}{a^2-b^2}+\frac {a^3 \int \frac {1-\tanh ^2(x)}{a+b \tanh (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}-\frac {\tanh (x)}{b}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tanh (x)\right )}{b^2 \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}+\frac {a^3 \log (a+b \tanh (x))}{b^2 \left (a^2-b^2\right )}-\frac {\tanh (x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 65, normalized size = 1.02 \[ \frac {\left (a b^2-a^3\right ) \log (\cosh (x))+a^3 \log (a \cosh (x)+b \sinh (x))+\left (b^3-a^2 b\right ) \tanh (x)-b^3 x}{b^2 (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.11, size = 264, normalized size = 4.12 \[ -\frac {{\left (a b^{2} + b^{3}\right )} x \cosh \relax (x)^{2} + 2 \, {\left (a b^{2} + b^{3}\right )} x \cosh \relax (x) \sinh \relax (x) + {\left (a b^{2} + b^{3}\right )} x \sinh \relax (x)^{2} - 2 \, a^{2} b + 2 \, b^{3} + {\left (a b^{2} + b^{3}\right )} x - {\left (a^{3} \cosh \relax (x)^{2} + 2 \, a^{3} \cosh \relax (x) \sinh \relax (x) + a^{3} \sinh \relax (x)^{2} + a^{3}\right )} \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{3} - a b^{2}\right )} \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 75, normalized size = 1.17 \[ \frac {a^{3} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} b^{2} - b^{4}} - \frac {x}{a - b} - \frac {a \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{2}} + \frac {2}{b {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 67, normalized size = 1.05 \[ -\frac {\tanh \relax (x )}{b}-\frac {\ln \left (\tanh \relax (x )-1\right )}{2 b +2 a}-\frac {\ln \left (1+\tanh \relax (x )\right )}{2 a -2 b}+\frac {a^{3} \ln \left (a +b \tanh \relax (x )\right )}{b^{2} \left (a +b \right ) \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 71, normalized size = 1.11 \[ \frac {a^{3} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} b^{2} - b^{4}} + \frac {x}{a + b} - \frac {a \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{2}} - \frac {2}{b e^{\left (-2 \, x\right )} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 59, normalized size = 0.92 \[ \frac {x}{a+b}-\frac {\mathrm {tanh}\relax (x)}{b}-\frac {a\,\ln \left (\mathrm {tanh}\relax (x)+1\right )}{a^2-b^2}+\frac {a^3\,\ln \left (a+b\,\mathrm {tanh}\relax (x)\right )}{b^2\,\left (a^2-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 330, normalized size = 5.16 \[ \begin {cases} \tilde {\infty } \left (x - \tanh {\relax (x )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {5 x \tanh {\relax (x )}}{2 b \tanh {\relax (x )} - 2 b} - \frac {5 x}{2 b \tanh {\relax (x )} - 2 b} - \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{2 b \tanh {\relax (x )} - 2 b} + \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )}}{2 b \tanh {\relax (x )} - 2 b} - \frac {2 \tanh ^{2}{\relax (x )}}{2 b \tanh {\relax (x )} - 2 b} + \frac {3}{2 b \tanh {\relax (x )} - 2 b} & \text {for}\: a = - b \\\frac {x \tanh {\relax (x )}}{2 b \tanh {\relax (x )} + 2 b} + \frac {x}{2 b \tanh {\relax (x )} + 2 b} + \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{2 b \tanh {\relax (x )} + 2 b} + \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )}}{2 b \tanh {\relax (x )} + 2 b} - \frac {2 \tanh ^{2}{\relax (x )}}{2 b \tanh {\relax (x )} + 2 b} + \frac {3}{2 b \tanh {\relax (x )} + 2 b} & \text {for}\: a = b \\\frac {x - \log {\left (\tanh {\relax (x )} + 1 \right )} - \frac {\tanh ^{2}{\relax (x )}}{2}}{a} & \text {for}\: b = 0 \\\frac {a^{3} \log {\left (\frac {a}{b} + \tanh {\relax (x )} \right )}}{a^{2} b^{2} - b^{4}} - \frac {a^{2} b \tanh {\relax (x )}}{a^{2} b^{2} - b^{4}} + \frac {a b^{2} x}{a^{2} b^{2} - b^{4}} - \frac {a b^{2} \log {\left (\tanh {\relax (x )} + 1 \right )}}{a^{2} b^{2} - b^{4}} - \frac {b^{3} x}{a^{2} b^{2} - b^{4}} + \frac {b^{3} \tanh {\relax (x )}}{a^{2} b^{2} - b^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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