Integrand size = 13, antiderivative size = 45 \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \arctan (\sinh (x))}{2 a}-\frac {2 \tanh (x)}{a}+\frac {3 \text {sech}(x) \tanh (x)}{2 a}-\frac {\text {sech}^2(x) \tanh (x)}{a+a \text {sech}(x)} \]
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Time = 0.06 (sec) , antiderivative size = 45, 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 = {3903, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \arctan (\sinh (x))}{2 a}-\frac {2 \tanh (x)}{a}-\frac {\tanh (x) \text {sech}^2(x)}{a \text {sech}(x)+a}+\frac {3 \tanh (x) \text {sech}(x)}{2 a} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 3903
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {sech}^2(x) \tanh (x)}{a+a \text {sech}(x)}-\frac {\int \text {sech}^2(x) (2 a-3 a \text {sech}(x)) \, dx}{a^2} \\ & = -\frac {\text {sech}^2(x) \tanh (x)}{a+a \text {sech}(x)}-\frac {2 \int \text {sech}^2(x) \, dx}{a}+\frac {3 \int \text {sech}^3(x) \, dx}{a} \\ & = \frac {3 \text {sech}(x) \tanh (x)}{2 a}-\frac {\text {sech}^2(x) \tanh (x)}{a+a \text {sech}(x)}-\frac {(2 i) \text {Subst}(\int 1 \, dx,x,-i \tanh (x))}{a}+\frac {3 \int \text {sech}(x) \, dx}{2 a} \\ & = \frac {3 \arctan (\sinh (x))}{2 a}-\frac {2 \tanh (x)}{a}+\frac {3 \text {sech}(x) \tanh (x)}{2 a}-\frac {\text {sech}^2(x) \tanh (x)}{a+a \text {sech}(x)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \arctan (\sinh (x))+2 \text {sech}^3(x) \tanh \left (\frac {x}{2}\right )-6 \tanh (x)+3 \text {sech}(x) \tanh (x)+2 \tanh ^3(x)}{2 a} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {-\tanh \left (\frac {x}{2}\right )+\frac {-3 \tanh \left (\frac {x}{2}\right )^{3}-\tanh \left (\frac {x}{2}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\) | \(46\) |
parallelrisch | \(\frac {-3 i \ln \left (-i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )+3 i \ln \left (i+\coth \left (x \right )-\operatorname {csch}\left (x \right )\right )+\left (-\operatorname {sech}\left (x \right )^{2}+2 \,\operatorname {sech}\left (x \right )+3\right ) \operatorname {csch}\left (x \right )-4 \coth \left (x \right )}{2 a}\) | \(52\) |
risch | \(\frac {3 \,{\mathrm e}^{4 x}+3 \,{\mathrm e}^{3 x}+5 \,{\mathrm e}^{2 x}+{\mathrm e}^{x}+4}{a \left (1+{\mathrm e}^{2 x}\right )^{2} \left ({\mathrm e}^{x}+1\right )}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{2 a}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{2 a}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (41) = 82\).
Time = 0.26 (sec) , antiderivative size = 325, normalized size of antiderivative = 7.22 \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, \cosh \left (x\right )^{4} + 3 \, {\left (4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{3} + {\left (18 \, \cosh \left (x\right )^{2} + 9 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{5} + {\left (5 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + \cosh \left (x\right )^{4} + 2 \, {\left (5 \, \cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 2 \, \cosh \left (x\right )^{3} + 2 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 5 \, \cosh \left (x\right )^{2} + {\left (12 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right ) + 4}{a \cosh \left (x\right )^{5} + a \sinh \left (x\right )^{5} + a \cosh \left (x\right )^{4} + {\left (5 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{3} + 2 \, {\left (5 \, a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{2} + a \cosh \left (x\right ) + {\left (5 \, a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} + 6 \, a \cosh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a} \]
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\[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {sech}^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
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Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.62 \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=-\frac {e^{\left (-x\right )} + 5 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 4}{a e^{\left (-x\right )} + 2 \, a e^{\left (-2 \, x\right )} + 2 \, a e^{\left (-3 \, x\right )} + a e^{\left (-4 \, x\right )} + a e^{\left (-5 \, x\right )} + a} - \frac {3 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, \arctan \left (e^{x}\right )}{a} + \frac {e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - e^{x} + 2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} + \frac {2}{a {\left (e^{x} + 1\right )}} \]
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Time = 2.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.62 \[ \int \frac {\text {sech}^4(x)}{a+a \text {sech}(x)} \, dx=\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {\frac {2}{a}+\frac {{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}}-\frac {2\,{\mathrm {e}}^x}{a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
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