Integrand size = 13, antiderivative size = 36 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}+\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}^3(x)}{3 a} \]
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Time = 0.04 (sec) , antiderivative size = 36, 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 = {3964, 76} \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=-\frac {\text {sech}^3(x)}{3 a}+\frac {\text {sech}^2(x)}{2 a}+\frac {\text {sech}(x)}{a}+\frac {\log (\cosh (x))}{a} \]
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Rule 76
Rule 3964
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a-a x)^2 (a+a x)}{x^4} \, dx,x,\cosh (x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {a^3}{x^3}-\frac {a^3}{x^2}+\frac {a^3}{x}\right ) \, dx,x,\cosh (x)\right )}{a^4} \\ & = \frac {\log (\cosh (x))}{a}+\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}^3(x)}{3 a} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {(2+6 \cosh (2 x)+3 \cosh (3 x) \log (\cosh (x))+\cosh (x) (6+9 \log (\cosh (x)))) \text {sech}^3(x)}{12 a} \]
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Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(-\frac {\frac {\operatorname {sech}\left (x \right )^{3}}{3}-\frac {\operatorname {sech}\left (x \right )^{2}}{2}-\operatorname {sech}\left (x \right )+\ln \left (\operatorname {sech}\left (x \right )\right )}{a}\) | \(26\) |
default | \(-\frac {\frac {\operatorname {sech}\left (x \right )^{3}}{3}-\frac {\operatorname {sech}\left (x \right )^{2}}{2}-\operatorname {sech}\left (x \right )+\ln \left (\operatorname {sech}\left (x \right )\right )}{a}\) | \(26\) |
risch | \(-\frac {x}{a}+\frac {2 \,{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+3 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}+3\right )}{3 \left (1+{\mathrm e}^{2 x}\right )^{3} a}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{a}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 437, normalized size of antiderivative = 12.14 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x \cosh \left (x\right )^{6} + 3 \, x \sinh \left (x\right )^{6} + 6 \, {\left (3 \, x \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{5} + 3 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{5} + 3 \, {\left (15 \, x \cosh \left (x\right )^{2} + 3 \, x - 10 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{4} + 4 \, {\left (15 \, x \cosh \left (x\right )^{3} + 3 \, {\left (3 \, x - 2\right )} \cosh \left (x\right ) - 15 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{3} + 3 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{3} + 3 \, {\left (15 \, x \cosh \left (x\right )^{4} + 6 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{2} - 20 \, \cosh \left (x\right )^{3} + 3 \, x - 4 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{2} - 3 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 6 \, {\left (3 \, x \cosh \left (x\right )^{5} + 2 \, {\left (3 \, x - 2\right )} \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )^{4} + {\left (3 \, x - 2\right )} \cosh \left (x\right ) - 2 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) + 3 \, x - 6 \, \cosh \left (x\right )}{3 \, {\left (a \cosh \left (x\right )^{6} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + a \sinh \left (x\right )^{6} + 3 \, a \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a \cosh \left (x\right )^{4} + 6 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a \cosh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )}} \]
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\[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\tanh ^{5}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.06 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {x}{a} + \frac {2 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, a e^{\left (-2 \, x\right )} + 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {11 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 12 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 12 \, e^{\left (-x\right )} - 12 \, e^{x} + 16}{6 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]
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Time = 2.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.67 \[ \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{a}-\frac {\frac {2}{a}+\frac {8\,{\mathrm {e}}^x}{3\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {\frac {2}{a}+\frac {2\,{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}+\frac {8\,{\mathrm {e}}^x}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )} \]
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