Optimal. Leaf size=21 \[ -\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5} \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 200}
\begin {gather*} -\frac {1}{5} \text {sech}^5(x)+\frac {2 \text {sech}^3(x)}{3}-\text {sech}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 2686
Rubi steps
\begin {align*} \int \text {sech}(x) \tanh ^5(x) \, dx &=-\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\text {sech}(x)\right )\\ &=-\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\text {sech}(x)+\frac {2 \text {sech}^3(x)}{3}-\frac {\text {sech}^5(x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 28, normalized size = 1.33
method | result | size |
default | \(-\frac {\sinh ^{4}\left (x \right )}{\cosh \left (x \right )^{5}}-\frac {4 \left (\sinh ^{2}\left (x \right )\right )}{3 \cosh \left (x \right )^{5}}-\frac {8}{15 \cosh \left (x \right )^{5}}\) | \(28\) |
risch | \(-\frac {2 \,{\mathrm e}^{x} \left (15 \,{\mathrm e}^{8 x}+20 \,{\mathrm e}^{6 x}+58 \,{\mathrm e}^{4 x}+20 \,{\mathrm e}^{2 x}+15\right )}{15 \left (1+{\mathrm e}^{2 x}\right )^{5}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 191 vs.
\(2 (17) = 34\).
time = 0.26, size = 191, normalized size = 9.10 \begin {gather*} -\frac {2 \, e^{\left (-x\right )}}{5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1} - \frac {8 \, e^{\left (-3 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {116 \, e^{\left (-5 \, x\right )}}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {8 \, e^{\left (-7 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1\right )}} - \frac {2 \, e^{\left (-9 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} + 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs.
\(2 (17) = 34\).
time = 0.34, size = 185, normalized size = 8.81 \begin {gather*} -\frac {2 \, {\left (15 \, \cosh \left (x\right )^{5} + 75 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 15 \, \sinh \left (x\right )^{5} + 5 \, {\left (30 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{3} + 35 \, \cosh \left (x\right )^{3} + 15 \, {\left (10 \, \cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (75 \, \cosh \left (x\right )^{4} + 15 \, \cosh \left (x\right )^{2} + 38\right )} \sinh \left (x\right ) + 78 \, \cosh \left (x\right )\right )}}{15 \, {\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} + 2\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 15 \, \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 8 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 10\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 29, normalized size = 1.38 \begin {gather*} - \frac {\tanh ^{4}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{5} - \frac {4 \tanh ^{2}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{15} - \frac {8 \operatorname {sech}{\left (x \right )}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (17) = 34\).
time = 0.40, size = 35, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 40 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 48\right )}}{15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 129, normalized size = 6.14 \begin {gather*} \frac {64\,{\mathrm {e}}^x}{5\,\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}{{\mathrm {e}}^{2\,x}+1}-\frac {176\,{\mathrm {e}}^x}{15\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {32\,{\mathrm {e}}^x}{5\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}+\frac {16\,{\mathrm {e}}^x}{3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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