Optimal. Leaf size=36 \[ \frac {x}{8}-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}-\frac {1}{8 (1+\coth (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3560, 8}
\begin {gather*} \frac {x}{8}-\frac {1}{8 (\coth (x)+1)}-\frac {1}{8 (\coth (x)+1)^2}-\frac {1}{6 (\coth (x)+1)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3560
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^3} \, dx &=-\frac {1}{6 (1+\coth (x))^3}+\frac {1}{2} \int \frac {1}{(1+\coth (x))^2} \, dx\\ &=-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}+\frac {1}{4} \int \frac {1}{1+\coth (x)} \, dx\\ &=-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}-\frac {1}{8 (1+\coth (x))}+\frac {\int 1 \, dx}{8}\\ &=\frac {x}{8}-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}-\frac {1}{8 (1+\coth (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 44, normalized size = 1.22 \begin {gather*} \frac {1}{96} (12 x+18 \cosh (2 x)-9 \cosh (4 x)+2 \cosh (6 x)-18 \sinh (2 x)+9 \sinh (4 x)-2 \sinh (6 x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.26, size = 40, normalized size = 1.11
method | result | size |
risch | \(\frac {x}{8}+\frac {3 \,{\mathrm e}^{-2 x}}{16}-\frac {3 \,{\mathrm e}^{-4 x}}{32}+\frac {{\mathrm e}^{-6 x}}{48}\) | \(23\) |
derivativedivides | \(-\frac {\ln \left (\coth \left (x \right )-1\right )}{16}-\frac {1}{6 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{8 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{8 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{16}\) | \(40\) |
default | \(-\frac {\ln \left (\coth \left (x \right )-1\right )}{16}-\frac {1}{6 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{8 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{8 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{16}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 22, normalized size = 0.61 \begin {gather*} \frac {1}{8} \, x + \frac {3}{16} \, e^{\left (-2 \, x\right )} - \frac {3}{32} \, e^{\left (-4 \, x\right )} + \frac {1}{48} \, e^{\left (-6 \, 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. 86 vs.
\(2 (28) = 56\).
time = 0.35, size = 86, normalized size = 2.39 \begin {gather*} \frac {2 \, {\left (6 \, x + 1\right )} \cosh \left (x\right )^{3} + 6 \, {\left (6 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 2 \, {\left (6 \, x - 1\right )} \sinh \left (x\right )^{3} + 3 \, {\left (2 \, {\left (6 \, x - 1\right )} \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right ) + 9 \, \cosh \left (x\right )}{96 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\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. 182 vs.
\(2 (31) = 62\).
time = 0.63, size = 182, normalized size = 5.06 \begin {gather*} \frac {3 x \tanh ^{3}{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} + \frac {9 x \tanh ^{2}{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} + \frac {9 x \tanh {\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} + \frac {3 x}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} + \frac {21 \tanh ^{2}{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} + \frac {27 \tanh {\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} + \frac {10}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh {\left (x \right )} + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 24, normalized size = 0.67 \begin {gather*} \frac {1}{96} \, {\left (18 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{8} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 22, normalized size = 0.61 \begin {gather*} \frac {x}{8}+\frac {3\,{\mathrm {e}}^{-2\,x}}{16}-\frac {3\,{\mathrm {e}}^{-4\,x}}{32}+\frac {{\mathrm {e}}^{-6\,x}}{48} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________