Integrand size = 13, antiderivative size = 79 \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=-\frac {b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}+\frac {\left (a^2-b^2\right ) \tanh (x)}{a^3}+\frac {b \tanh ^2(x)}{2 a^2}-\frac {\tanh ^3(x)}{3 a} \]
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Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3597, 908} \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=\frac {b \tanh ^2(x)}{2 a^2}-\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}+\frac {\left (a^2-b^2\right ) \tanh (x)}{a^3}-\frac {\tanh ^3(x)}{3 a} \]
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Rule 908
Rule 3597
Rubi steps \begin{align*} \text {integral}& = -\left (b \text {Subst}\left (\int \frac {-b^2+x^2}{x^4 (a+x)} \, dx,x,b \coth (x)\right )\right ) \\ & = -\left (b \text {Subst}\left (\int \left (-\frac {b^2}{a x^4}+\frac {b^2}{a^2 x^3}+\frac {a^2-b^2}{a^3 x^2}+\frac {-a^2+b^2}{a^4 x}+\frac {a^2-b^2}{a^4 (a+x)}\right ) \, dx,x,b \coth (x)\right )\right ) \\ & = -\frac {b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}+\frac {\left (a^2-b^2\right ) \tanh (x)}{a^3}+\frac {b \tanh ^2(x)}{2 a^2}-\frac {\tanh ^3(x)}{3 a} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76 \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=\frac {6 b \left (-a^2+b^2\right ) \log (b+a \tanh (x))+6 a \left (a^2-b^2\right ) \tanh (x)+3 a^2 b \tanh ^2(x)-2 a^3 \tanh ^3(x)}{6 a^4} \]
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Time = 11.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.84
method | result | size |
risch | \(-\frac {2 \left (3 a b \,{\mathrm e}^{4 x}-3 b^{2} {\mathrm e}^{4 x}+6 a^{2} {\mathrm e}^{2 x}+3 b \,{\mathrm e}^{2 x} a -6 b^{2} {\mathrm e}^{2 x}+2 a^{2}-3 b^{2}\right )}{3 a^{3} \left (1+{\mathrm e}^{2 x}\right )^{3}}+\frac {b \ln \left (1+{\mathrm e}^{2 x}\right )}{a^{2}}-\frac {b^{3} \ln \left (1+{\mathrm e}^{2 x}\right )}{a^{4}}-\frac {b \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{4}}\) | \(145\) |
default | \(-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{a^{4}}-\frac {2 \left (\frac {\left (-a^{3}+a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{5}-a^{2} b \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {2}{3} a^{3}+2 a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}-a^{2} b \tanh \left (\frac {x}{2}\right )^{2}+\left (-a^{3}+a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{2}\right )}{a^{4}}\) | \(151\) |
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Leaf count of result is larger than twice the leaf count of optimal. 909 vs. \(2 (75) = 150\).
Time = 0.29 (sec) , antiderivative size = 909, normalized size of antiderivative = 11.51 \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=\int \frac {\operatorname {sech}^{4}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.68 \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=\frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2} + 3 \, {\left (2 \, a^{2} - a b - 2 \, b^{2}\right )} e^{\left (-2 \, x\right )} - 3 \, {\left (a b + b^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, {\left (3 \, a^{3} e^{\left (-2 \, x\right )} + 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} + a^{3}\right )}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{4}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (75) = 150\).
Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.54 \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=-\frac {{\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{5} + a^{4} b} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{a^{4}} - \frac {11 \, a^{2} b e^{\left (6 \, x\right )} - 11 \, b^{3} e^{\left (6 \, x\right )} + 45 \, a^{2} b e^{\left (4 \, x\right )} - 12 \, a b^{2} e^{\left (4 \, x\right )} - 33 \, b^{3} e^{\left (4 \, x\right )} + 24 \, a^{3} e^{\left (2 \, x\right )} + 45 \, a^{2} b e^{\left (2 \, x\right )} - 24 \, a b^{2} e^{\left (2 \, x\right )} - 33 \, b^{3} e^{\left (2 \, x\right )} + 8 \, a^{3} + 11 \, a^{2} b - 12 \, a b^{2} - 11 \, b^{3}}{6 \, a^{4} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
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Time = 2.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.56 \[ \int \frac {\text {sech}^4(x)}{a+b \coth (x)} \, dx=\frac {8}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {2\,\left (2\,a-b\right )}{a^2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}-\frac {2\,b\,\left (a-b\right )}{a^3\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {b\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a+b\right )\,\left (a-b\right )}{a^4}+\frac {b\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\left (a+b\right )\,\left (a-b\right )}{a^4} \]
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