Integrand size = 13, antiderivative size = 76 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=\frac {a x}{a^2-b^2}-\frac {b \log (\cosh (x))}{a^2-b^2}-\frac {a^4 \log (a+b \tanh (x))}{b^3 \left (a^2-b^2\right )}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]
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Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3647, 3728, 3708, 3698, 31, 3556} \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=\frac {a x}{a^2-b^2}-\frac {b \log (\cosh (x))}{a^2-b^2}-\frac {a^4 \log (a+b \tanh (x))}{b^3 \left (a^2-b^2\right )}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]
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Rule 31
Rule 3556
Rule 3647
Rule 3698
Rule 3708
Rule 3728
Rubi steps \begin{align*} \text {integral}& = -\frac {\tanh ^2(x)}{2 b}-\frac {\int \frac {\tanh (x) \left (-2 a-2 b \tanh (x)+2 a \tanh ^2(x)\right )}{a+b \tanh (x)} \, dx}{2 b} \\ & = \frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b}-\frac {\int \frac {2 a^2-2 \left (a^2+b^2\right ) \tanh ^2(x)}{a+b \tanh (x)} \, dx}{2 b^2} \\ & = \frac {a x}{a^2-b^2}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b}-\frac {a^4 \int \frac {1-\tanh ^2(x)}{a+b \tanh (x)} \, dx}{b^2 \left (a^2-b^2\right )}-\frac {b \int \tanh (x) \, dx}{a^2-b^2} \\ & = \frac {a x}{a^2-b^2}-\frac {b \log (\cosh (x))}{a^2-b^2}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b}-\frac {a^4 \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tanh (x)\right )}{b^3 \left (a^2-b^2\right )} \\ & = \frac {a x}{a^2-b^2}-\frac {b \log (\cosh (x))}{a^2-b^2}-\frac {a^4 \log (a+b \tanh (x))}{b^3 \left (a^2-b^2\right )}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=-\frac {\log (1-\tanh (x))}{2 (a+b)}+\frac {\log (1+\tanh (x))}{2 (a-b)}-\frac {a^4 \log (a+b \tanh (x))}{b^3 \left (a^2-b^2\right )}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]
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Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {\tanh \left (x \right )^{2}}{2 b}+\frac {a \tanh \left (x \right )}{b^{2}}-\frac {a^{4} \ln \left (a +b \tanh \left (x \right )\right )}{b^{3} \left (a +b \right ) \left (a -b \right )}+\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 a +2 b}\) | \(76\) |
default | \(-\frac {\tanh \left (x \right )^{2}}{2 b}+\frac {a \tanh \left (x \right )}{b^{2}}-\frac {a^{4} \ln \left (a +b \tanh \left (x \right )\right )}{b^{3} \left (a +b \right ) \left (a -b \right )}+\frac {\ln \left (1+\tanh \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{2 a +2 b}\) | \(76\) |
parallelrisch | \(-\frac {\tanh \left (x \right )^{2} a^{2} b^{2}-\tanh \left (x \right )^{2} b^{4}-2 \ln \left (1-\tanh \left (x \right )\right ) b^{4}+2 a^{4} \ln \left (a +b \tanh \left (x \right )\right )-2 b^{3} a x -2 b^{4} x -2 b \tanh \left (x \right ) a^{3}+2 \tanh \left (x \right ) a \,b^{3}}{2 b^{3} \left (a^{2}-b^{2}\right )}\) | \(91\) |
risch | \(\frac {x}{a +b}-\frac {2 x \,a^{2}}{b^{3}}-\frac {2 x}{b}+\frac {2 x \,a^{4}}{b^{3} \left (a^{2}-b^{2}\right )}-\frac {2 \left (a \,{\mathrm e}^{2 x}-b \,{\mathrm e}^{2 x}+a \right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} b^{2}}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right ) a^{2}}{b^{3}}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{b}-\frac {a^{4} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{b^{3} \left (a^{2}-b^{2}\right )}\) | \(133\) |
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Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 644, normalized size of antiderivative = 8.47 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=\frac {{\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right )^{4} + 4 \, {\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a b^{3} + b^{4}\right )} x \sinh \left (x\right )^{4} - 2 \, a^{3} b + 2 \, a b^{3} - 2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a b^{3} + b^{4}\right )} x\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - 3 \, {\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right )^{2} - {\left (a b^{3} + b^{4}\right )} x\right )} \sinh \left (x\right )^{2} + {\left (a b^{3} + b^{4}\right )} x - {\left (a^{4} \cosh \left (x\right )^{4} + 4 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{4} \sinh \left (x\right )^{4} + 2 \, a^{4} \cosh \left (x\right )^{2} + a^{4} + 2 \, {\left (3 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{4} \cosh \left (x\right )^{3} + a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left ({\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} - b^{4}\right )} \sinh \left (x\right )^{4} + a^{4} - b^{4} + 2 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} - b^{4} + 3 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{3} + {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left ({\left (a b^{3} + b^{4}\right )} x \cosh \left (x\right )^{3} - {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a b^{3} + b^{4}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b^{3} - b^{5} + 3 \, {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (61) = 122\).
Time = 0.39 (sec) , antiderivative size = 442, normalized size of antiderivative = 5.82 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=\begin {cases} \tilde {\infty } \left (x - \log {\left (\tanh {\left (x \right )} + 1 \right )} - \frac {\tanh ^{2}{\left (x \right )}}{2}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {x - \frac {\tanh ^{3}{\left (x \right )}}{3} - \tanh {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {7 x \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} - 2 b} - \frac {7 x}{2 b \tanh {\left (x \right )} - 2 b} - \frac {4 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} - 2 b} + \frac {4 \log {\left (\tanh {\left (x \right )} + 1 \right )}}{2 b \tanh {\left (x \right )} - 2 b} - \frac {\tanh ^{3}{\left (x \right )}}{2 b \tanh {\left (x \right )} - 2 b} - \frac {\tanh ^{2}{\left (x \right )}}{2 b \tanh {\left (x \right )} - 2 b} + \frac {3}{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 {4 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{2 b \tanh {\left (x \right )} + 2 b} - \frac {4 \log {\left (\tanh {\left (x \right )} + 1 \right )}}{2 b \tanh {\left (x \right )} + 2 b} - \frac {\tanh ^{3}{\left (x \right )}}{2 b \tanh {\left (x \right )} + 2 b} + \frac {\tanh ^{2}{\left (x \right )}}{2 b \tanh {\left (x \right )} + 2 b} - \frac {3}{2 b \tanh {\left (x \right )} + 2 b} & \text {for}\: a = b \\- \frac {2 a^{4} \log {\left (\frac {a}{b} + \tanh {\left (x \right )} \right )}}{2 a^{2} b^{3} - 2 b^{5}} + \frac {2 a^{3} b \tanh {\left (x \right )}}{2 a^{2} b^{3} - 2 b^{5}} - \frac {a^{2} b^{2} \tanh ^{2}{\left (x \right )}}{2 a^{2} b^{3} - 2 b^{5}} + \frac {2 a b^{3} x}{2 a^{2} b^{3} - 2 b^{5}} - \frac {2 a b^{3} \tanh {\left (x \right )}}{2 a^{2} b^{3} - 2 b^{5}} - \frac {2 b^{4} x}{2 a^{2} b^{3} - 2 b^{5}} + \frac {2 b^{4} \log {\left (\tanh {\left (x \right )} + 1 \right )}}{2 a^{2} b^{3} - 2 b^{5}} + \frac {b^{4} \tanh ^{2}{\left (x \right )}}{2 a^{2} b^{3} - 2 b^{5}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.32 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=-\frac {a^{4} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} b^{3} - b^{5}} + \frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} + a\right )}}{2 \, b^{2} e^{\left (-2 \, x\right )} + b^{2} e^{\left (-4 \, x\right )} + b^{2}} + \frac {x}{a + b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.29 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=-\frac {a^{4} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} b^{3} - b^{5}} + \frac {x}{a - b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{3}} - \frac {2 \, {\left (a b + {\left (a b - b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{b^{3} {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
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Time = 1.76 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {\tanh ^4(x)}{a+b \tanh (x)} \, dx=\frac {x}{a+b}-\frac {{\mathrm {tanh}\left (x\right )}^2}{2\,b}+\frac {b\,\ln \left (\mathrm {tanh}\left (x\right )+1\right )}{a^2-b^2}+\frac {a\,\mathrm {tanh}\left (x\right )}{b^2}-\frac {a^4\,\ln \left (a+b\,\mathrm {tanh}\left (x\right )\right )}{b^3\,\left (a^2-b^2\right )} \]
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