Optimal. Leaf size=38 \[ \frac {x}{2}-\frac {2 \text {ArcTan}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6 (1+\tanh (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2083, 213, 632,
210} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {x}{2}-\frac {1}{6 (\tanh (x)+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 213
Rule 632
Rule 2083
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x)}{\cosh ^3(x)+\sinh ^3(x)} \, dx &=\text {Subst}\left (\int \frac {1}{1-x^2+x^3-x^5} \, dx,x,\tanh (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{6 (1+x)^2}-\frac {1}{2 \left (-1+x^2\right )}+\frac {1}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{6 (1+\tanh (x))}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tanh (x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {x}{2}-\frac {1}{6 (1+\tanh (x))}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tanh (x)\right )\\ &=\frac {x}{2}-\frac {2 \tan ^{-1}\left (\frac {1-2 \tanh (x)}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6 (1+\tanh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 40, normalized size = 1.05 \begin {gather*} \frac {1}{36} \left (18 x+8 \sqrt {3} \text {ArcTan}\left (\frac {-1+2 \tanh (x)}{\sqrt {3}}\right )-3 \cosh (2 x)+3 \sinh (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 1.92, size = 96, normalized size = 2.53
method | result | size |
risch | \(\frac {x}{2}-\frac {{\mathrm e}^{-2 x}}{12}+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {3}\right )}{9}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {3}\right )}{9}\) | \(47\) |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (-i \sqrt {3}-1\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+\left (i \sqrt {3}-1\right ) \tanh \left (\frac {x}{2}\right )+1\right )}{9}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{3 \tanh \left (\frac {x}{2}\right )+3}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (29) = 58\).
time = 0.48, size = 73, normalized size = 1.92 \begin {gather*} \frac {2}{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 {2}{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}{2} \, x - \frac {1}{12} \, e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (29) = 58\).
time = 0.41, size = 95, normalized size = 2.50 \begin {gather*} \frac {18 \, x \cosh \left (x\right )^{2} + 36 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 18 \, x \sinh \left (x\right )^{2} - 8 \, {\left (\sqrt {3} \cosh \left (x\right )^{2} + 2 \, \sqrt {3} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {3} \sinh \left (x\right )^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) - 3}{36 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (36) = 72\).
time = 0.55, size = 136, normalized size = 3.58 \begin {gather*} \frac {9 x \sinh {\left (x \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} + \frac {9 x \cosh {\left (x \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} + \frac {4 \sqrt {3} \sinh {\left (x \right )} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \sqrt {3} \cosh {\left (x \right )}}{3 \sinh {\left (x \right )}} \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} + \frac {4 \sqrt {3} \cosh {\left (x \right )} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \sqrt {3} \cosh {\left (x \right )}}{3 \sinh {\left (x \right )}} \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} - \frac {3 \cosh {\left (x \right )}}{18 \sinh {\left (x \right )} + 18 \cosh {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 33, normalized size = 0.87 \begin {gather*} -\frac {1}{12} \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, x\right )}\right ) + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.69, size = 25, normalized size = 0.66 \begin {gather*} \frac {x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{12}+\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^{2\,x}}{3}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________