Optimal. Leaf size=16 \[ -\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4442, 213}
\begin {gather*} -\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 4442
Rubi steps
\begin {align*} \int \text {sech}(2 x) \sinh (x) \, dx &=\text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 155, normalized size = 9.69 \begin {gather*} \frac {-2 i \text {ArcTan}\left (\frac {\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (-1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )+2 i \text {ArcTan}\left (\frac {\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}{\left (-1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )-4 \tanh ^{-1}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )+\log \left (\sqrt {2}-2 \cosh (x)\right )-\log \left (\sqrt {2}+2 \cosh (x)\right )}{4 \sqrt {2}} \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. \(38\) vs.
\(2(12)=24\).
time = 0.89, size = 39, normalized size = 2.44
method | result | size |
risch | \(\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{4}-\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{4}\) | \(39\) |
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. 42 vs.
\(2 (12) = 24\).
time = 0.47, size = 42, normalized size = 2.62 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, 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. 35 vs.
\(2 (12) = 24\).
time = 0.38, size = 35, normalized size = 2.19 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt {2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\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 \sinh {\left (x \right )} \operatorname {sech}{\left (2 x \right )}\, dx \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. 38 vs.
\(2 (12) = 24\).
time = 0.38, size = 38, normalized size = 2.38 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 35, normalized size = 2.19 \begin {gather*} -\frac {\sqrt {2}\,\left (\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )-\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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