Optimal. Leaf size=33 \[ -\frac {1}{3 (\tanh (x)+1)}-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.14, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2074, 618, 204} \[ -\frac {1}{3 (\tanh (x)+1)}-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2074
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1+x+x^2}{1+x+x^3+x^4} \, dx,x,\tanh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{3 (1+x)^2}+\frac {2}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{3 (1+\tanh (x))}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{3 (1+\tanh (x))}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right )\\ &=-\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3 (1+\tanh (x))}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 37, normalized size = 1.12 \[ \frac {1}{18} \left (3 \sinh (2 x)-3 \cosh (2 x)+8 \sqrt {3} \tan ^{-1}\left (\frac {2 \tanh (x)-1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^3(x)-\sinh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.30, size = 74, normalized size = 2.24 \[ -\frac {8 \, {\left (\sqrt {3} \cosh \relax (x)^{2} + 2 \, \sqrt {3} \cosh \relax (x) \sinh \relax (x) + \sqrt {3} \sinh \relax (x)^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) + 3}{18 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 22, normalized size = 0.67 \[ \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, x\right )}\right ) - \frac {1}{6} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 44, normalized size = 1.33
method | result | size |
risch | \(-\frac {{\mathrm e}^{-2 x}}{6}+\frac {2 i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {3}\right )}{9}-\frac {2 i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {3}\right )}{9}\) | \(44\) |
default | \(\frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-1-i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-1+i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {2}{3 \left (1+\tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {2}{3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}\) | \(78\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 70, normalized size = 2.12 \[ \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-x\right )} - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {1}{6} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 22, normalized size = 0.67 \[ \frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^{2\,x}}{3}\right )}{9}-\frac {{\mathrm {e}}^{-2\,x}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.83, size = 102, normalized size = 3.09 \[ \frac {4 \sqrt {3} \sinh {\relax (x )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sinh {\relax (x )}}{3 \cosh {\relax (x )}} - \frac {\sqrt {3}}{3} \right )}}{9 \sinh {\relax (x )} + 9 \cosh {\relax (x )}} + \frac {3 \sinh {\relax (x )}}{9 \sinh {\relax (x )} + 9 \cosh {\relax (x )}} + \frac {4 \sqrt {3} \cosh {\relax (x )} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sinh {\relax (x )}}{3 \cosh {\relax (x )}} - \frac {\sqrt {3}}{3} \right )}}{9 \sinh {\relax (x )} + 9 \cosh {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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