Optimal. Leaf size=17 \[ \frac {\tanh ^2(x)}{2}-\frac {\tanh ^3(x)}{3} \]
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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3597, 862, 45}
\begin {gather*} \frac {\tanh ^2(x)}{2}-\frac {\tanh ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 862
Rule 3597
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(x)}{1+\coth (x)} \, dx &=-\text {Subst}\left (\int \frac {-1+x^2}{x^4 (1+x)} \, dx,x,\coth (x)\right )\\ &=-\text {Subst}\left (\int \frac {-1+x}{x^4} \, dx,x,\coth (x)\right )\\ &=-\text {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {1}{x^3}\right ) \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^2(x)}{2}-\frac {\tanh ^3(x)}{3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{6} \left (-3 \text {sech}^2(x)-2 \tanh ^3(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(37\) vs.
\(2(13)=26\).
time = 0.59, size = 38, normalized size = 2.24
method | result | size |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{2 x}-1\right )}{3 \left (1+{\mathrm e}^{2 x}\right )^{3}}\) | \(19\) |
default | \(-\frac {4 \left (-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{2}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}\) | \(38\) |
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. 75 vs.
\(2 (13) = 26\).
time = 0.26, size = 75, normalized size = 4.41 \begin {gather*} -\frac {2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac {4 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac {2}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \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. 84 vs.
\(2 (13) = 26\).
time = 0.36, size = 84, normalized size = 4.94 \begin {gather*} -\frac {4 \, {\left (\cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right )}}{3 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + {\left (10 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{3} + {\left (10 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {sech}^{4}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 18, normalized size = 1.06 \begin {gather*} -\frac {2\,\left (3\,{\mathrm {e}}^{2\,x}-1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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