Optimal. Leaf size=16 \[ -\frac {2}{1+\cosh (x)}-\log (1+\cosh (x)) \]
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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4477, 2746, 45}
\begin {gather*} -\frac {2}{\cosh (x)+1}-\log (\cosh (x)+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rule 4477
Rubi steps
\begin {align*} \int (-\coth (x)+\text {csch}(x))^3 \, dx &=i \int (i-i \cosh (x))^3 \text {csch}^3(x) \, dx\\ &=-\text {Subst}\left (\int \frac {i+x}{(i-x)^2} \, dx,x,-i \cosh (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {2 i}{(-i+x)^2}+\frac {1}{-i+x}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac {2 i}{i+i \cosh (x)}-\log (1+\cosh (x))\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(16)=32\).
time = 0.04, size = 43, normalized size = 2.69 \begin {gather*} 2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (\sinh \left (\frac {x}{2}\right )\right )-\log (\sinh (x))+3 \log \left (\tanh \left (\frac {x}{2}\right )\right )-\text {sech}^2\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.85, size = 20, normalized size = 1.25
method | result | size |
risch | \(x -\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right )^{2}}-2 \ln \left ({\mathrm e}^{x}+1\right )\) | \(20\) |
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. 68 vs.
\(2 (16) = 32\).
time = 0.26, size = 68, normalized size = 4.25 \begin {gather*} \frac {3}{2} \, \coth \left (x\right )^{2} - x + \frac {4 \, {\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - 2 \, \log \left (e^{\left (-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. 88 vs.
\(2 (16) = 32\).
time = 0.41, size = 88, normalized size = 5.50 \begin {gather*} \frac {x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} + 2 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \, {\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \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 3 \coth {\left (x \right )} \operatorname {csch}^{2}{\left (x \right )}\, dx - \int \left (- 3 \coth ^{2}{\left (x \right )} \operatorname {csch}{\left (x \right )}\right )\, dx - \int \coth ^{3}{\left (x \right )}\, dx - \int \left (- \operatorname {csch}^{3}{\left (x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 19, normalized size = 1.19 \begin {gather*} x - \frac {4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} - 2 \, \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.54, size = 31, normalized size = 1.94 \begin {gather*} x-2\,\ln \left ({\mathrm {e}}^x+1\right )+\frac {4}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {4}{{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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