Integrand size = 13, antiderivative size = 34 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\coth ^3(x)}{3 a}-\frac {\coth ^5(x)}{5 a}+\frac {\text {csch}^5(x)}{5 a} \]
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Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3957, 2918, 2686, 30, 2687, 14} \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {\coth ^5(x)}{5 a}+\frac {\coth ^3(x)}{3 a}+\frac {\text {csch}^5(x)}{5 a} \]
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Rule 14
Rule 30
Rule 2686
Rule 2687
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\coth (x) \text {csch}^3(x)}{-a-a \cosh (x)} \, dx \\ & = \frac {\int \coth ^2(x) \text {csch}^4(x) \, dx}{a}-\frac {\int \coth (x) \text {csch}^5(x) \, dx}{a} \\ & = \frac {i \text {Subst}\left (\int x^4 \, dx,x,-i \text {csch}(x)\right )}{a}+\frac {i \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \coth (x)\right )}{a} \\ & = \frac {\text {csch}^5(x)}{5 a}+\frac {i \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \coth (x)\right )}{a} \\ & = \frac {\coth ^3(x)}{3 a}-\frac {\coth ^5(x)}{5 a}+\frac {\text {csch}^5(x)}{5 a} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {(-15-6 \cosh (x)-2 \cosh (2 x)+2 \cosh (3 x)+\cosh (4 x)) \text {csch}^3(x)}{60 a (1+\cosh (x))} \]
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Time = 0.89 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{5}}{5}+\frac {2 \tanh \left (\frac {x}{2}\right )^{3}}{3}+\frac {2}{\tanh \left (\frac {x}{2}\right )}-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}}{16 a}\) | \(39\) |
risch | \(-\frac {4 \left (15 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x}-1\right )}{15 \left ({\mathrm e}^{x}+1\right )^{5} a \left ({\mathrm e}^{x}-1\right )^{3}}\) | \(42\) |
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (28) = 56\).
Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 6.44 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {8 \, {\left (7 \, \cosh \left (x\right )^{2} + 4 \, {\left (4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 7 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )}}{15 \, {\left (a \cosh \left (x\right )^{6} + a \sinh \left (x\right )^{6} + 2 \, a \cosh \left (x\right )^{5} + 2 \, {\left (3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{5} - 2 \, a \cosh \left (x\right )^{4} + {\left (15 \, a \cosh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) - 2 \, a\right )} \sinh \left (x\right )^{4} - 6 \, a \cosh \left (x\right )^{3} + 2 \, {\left (10 \, a \cosh \left (x\right )^{3} + 10 \, a \cosh \left (x\right )^{2} - 4 \, a \cosh \left (x\right ) - 3 \, a\right )} \sinh \left (x\right )^{3} - a \cosh \left (x\right )^{2} + {\left (15 \, a \cosh \left (x\right )^{4} + 20 \, a \cosh \left (x\right )^{3} - 12 \, a \cosh \left (x\right )^{2} - 18 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{4} - 4 \, a \cosh \left (x\right )^{3} - 9 \, a \cosh \left (x\right )^{2} + a \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right ) + 2 \, a\right )}} \]
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\[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (28) = 56\).
Time = 0.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 8.59 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {8 \, e^{\left (-x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {8 \, e^{\left (-2 \, x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {8 \, e^{\left (-3 \, x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {4 \, e^{\left (-4 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a} + \frac {4}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 5}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {15 \, e^{\left (4 \, x\right )} + 60 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 20 \, e^{x} + 7}{120 \, a {\left (e^{x} + 1\right )}^{5}} \]
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Time = 1.99 (sec) , antiderivative size = 236, normalized size of antiderivative = 6.94 \[ \int \frac {\text {csch}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{40\,a}-\frac {{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^{2\,x}}{40\,a}-\frac {1}{24\,a}+\frac {{\mathrm {e}}^x}{20\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^{3\,x}}{10\,a}-\frac {{\mathrm {e}}^{2\,x}}{4\,a}+\frac {{\mathrm {e}}^{4\,x}}{40\,a}+\frac {1}{40\,a}+\frac {{\mathrm {e}}^x}{10\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{20\,a\,\left ({\mathrm {e}}^x+1\right )} \]
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