Optimal. Leaf size=35 \[ \frac {3 \tan ^{-1}(\sinh (x))}{8 a}+\frac {\tanh (x) \text {sech}^3(x)}{4 a}+\frac {3 \tanh (x) \text {sech}(x)}{8 a} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3175, 3768, 3770} \[ \frac {3 \tan ^{-1}(\sinh (x))}{8 a}+\frac {\tanh (x) \text {sech}^3(x)}{4 a}+\frac {3 \tanh (x) \text {sech}(x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(x)}{a+a \sinh ^2(x)} \, dx &=\frac {\int \text {sech}^5(x) \, dx}{a}\\ &=\frac {\text {sech}^3(x) \tanh (x)}{4 a}+\frac {3 \int \text {sech}^3(x) \, dx}{4 a}\\ &=\frac {3 \text {sech}(x) \tanh (x)}{8 a}+\frac {\text {sech}^3(x) \tanh (x)}{4 a}+\frac {3 \int \text {sech}(x) \, dx}{8 a}\\ &=\frac {3 \tan ^{-1}(\sinh (x))}{8 a}+\frac {3 \text {sech}(x) \tanh (x)}{8 a}+\frac {\text {sech}^3(x) \tanh (x)}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 0.97 \[ \frac {\frac {3}{4} \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{4} \tanh (x) \text {sech}^3(x)+\frac {3}{8} \tanh (x) \text {sech}(x)}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.55, size = 488, normalized size = 13.94 \[ \frac {3 \, \cosh \relax (x)^{7} + 21 \, \cosh \relax (x) \sinh \relax (x)^{6} + 3 \, \sinh \relax (x)^{7} + {\left (63 \, \cosh \relax (x)^{2} + 11\right )} \sinh \relax (x)^{5} + 11 \, \cosh \relax (x)^{5} + 5 \, {\left (21 \, \cosh \relax (x)^{3} + 11 \, \cosh \relax (x)\right )} \sinh \relax (x)^{4} + {\left (105 \, \cosh \relax (x)^{4} + 110 \, \cosh \relax (x)^{2} - 11\right )} \sinh \relax (x)^{3} - 11 \, \cosh \relax (x)^{3} + {\left (63 \, \cosh \relax (x)^{5} + 110 \, \cosh \relax (x)^{3} - 33 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 3 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 4 \, {\left (7 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{6} + 4 \, \cosh \relax (x)^{6} + 8 \, {\left (7 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} + 30 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} + 10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (7 \, \cosh \relax (x)^{6} + 15 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 8 \, {\left (\cosh \relax (x)^{7} + 3 \, \cosh \relax (x)^{5} + 3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + {\left (21 \, \cosh \relax (x)^{6} + 55 \, \cosh \relax (x)^{4} - 33 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x) - 3 \, \cosh \relax (x)}{4 \, {\left (a \cosh \relax (x)^{8} + 8 \, a \cosh \relax (x) \sinh \relax (x)^{7} + a \sinh \relax (x)^{8} + 4 \, a \cosh \relax (x)^{6} + 4 \, {\left (7 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)^{6} + 8 \, {\left (7 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 6 \, a \cosh \relax (x)^{4} + 2 \, {\left (35 \, a \cosh \relax (x)^{4} + 30 \, a \cosh \relax (x)^{2} + 3 \, a\right )} \sinh \relax (x)^{4} + 8 \, {\left (7 \, a \cosh \relax (x)^{5} + 10 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, a \cosh \relax (x)^{2} + 4 \, {\left (7 \, a \cosh \relax (x)^{6} + 15 \, a \cosh \relax (x)^{4} + 9 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)^{2} + 8 \, {\left (a \cosh \relax (x)^{7} + 3 \, a \cosh \relax (x)^{5} + 3 \, a \cosh \relax (x)^{3} + a \cosh \relax (x)\right )} \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 67, normalized size = 1.91 \[ \frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )}}{16 \, a} - \frac {3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}}{4 \, {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 94, normalized size = 2.69 \[ -\frac {5 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {3 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {3 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {5 \tanh \left (\frac {x}{2}\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 69, normalized size = 1.97 \[ \frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} - \frac {3 \, \arctan \left (e^{\left (-x\right )}\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 118, normalized size = 3.37 \[ \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{4\,\sqrt {a^2}}-\frac {4\,{\mathrm {e}}^{3\,x}}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,{\mathrm {e}}^x}{a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {{\mathrm {e}}^x}{2\,a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {3\,{\mathrm {e}}^x}{4\,a\,\left ({\mathrm {e}}^{2\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {sech}^{3}{\relax (x )}}{\sinh ^{2}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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