Optimal. Leaf size=20 \[ -\frac {1}{2} \text {ArcTan}(\sinh (x))-\text {sech}(x)+\frac {1}{2} \text {sech}(x) \tanh (x) \]
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Rubi [A]
time = 0.11, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3599, 3187,
3186, 2686, 8, 2691, 3855} \begin {gather*} -\frac {1}{2} \text {ArcTan}(\sinh (x))-\text {sech}(x)+\frac {1}{2} \tanh (x) \text {sech}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2686
Rule 2691
Rule 3186
Rule 3187
Rule 3599
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(x)}{1+\coth (x)} \, dx &=-\left (i \int \frac {\text {sech}^2(x) \tanh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int \text {sech}^2(x) (-\cosh (x)+\sinh (x)) \tanh (x) \, dx\\ &=i \int \left (-i \text {sech}(x) \tanh (x)+i \text {sech}(x) \tanh ^2(x)\right ) \, dx\\ &=\int \text {sech}(x) \tanh (x) \, dx-\int \text {sech}(x) \tanh ^2(x) \, dx\\ &=\frac {1}{2} \text {sech}(x) \tanh (x)-\frac {1}{2} \int \text {sech}(x) \, dx-\text {Subst}(\int 1 \, dx,x,\text {sech}(x))\\ &=-\frac {1}{2} \tan ^{-1}(\sinh (x))-\text {sech}(x)+\frac {1}{2} \text {sech}(x) \tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} -\text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{2} \text {sech}(x) (-2+\tanh (x)) \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. \(44\) vs.
\(2(16)=32\).
time = 0.59, size = 45, normalized size = 2.25
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x}+3\right )}{\left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}\) | \(38\) |
default | \(\frac {-\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+\tanh \left (\frac {x}{2}\right )-2}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\arctan \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(45\) |
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. 33 vs.
\(2 (16) = 32\).
time = 0.48, size = 33, normalized size = 1.65 \begin {gather*} -\frac {e^{\left (-x\right )} + 3 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + \arctan \left (e^{\left (-x\right )}\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. 140 vs.
\(2 (16) = 32\).
time = 0.35, size = 140, normalized size = 7.00 \begin {gather*} -\frac {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 3 \, {\left (\cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \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}^{3}{\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 = 25, normalized size = 1.25 \begin {gather*} -\frac {e^{\left (3 \, x\right )} + 3 \, e^{x}}{{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} - \arctan \left (e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 22, normalized size = 1.10 \begin {gather*} -\mathrm {atan}\left ({\mathrm {e}}^x\right )-\frac {1}{2\,\mathrm {cosh}\left (x\right )}-\frac {{\mathrm {e}}^{-x}}{2\,{\mathrm {cosh}\left (x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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