Optimal. Leaf size=37 \[ -\frac {3 x}{2}+\frac {3 \coth (x)}{2}-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+2 \log (\sinh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3631, 3609,
3606, 3556} \begin {gather*} -\frac {3 x}{2}+\frac {\coth ^3(x)}{2 (\coth (x)+1)}-\coth ^2(x)+\frac {3 \coth (x)}{2}+2 \log (\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rule 3606
Rule 3609
Rule 3631
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{1+\coth (x)} \, dx &=\frac {\coth ^3(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (3-4 \coth (x)) \coth ^2(x) \, dx\\ &=-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+\frac {1}{2} i \int (-4 i+3 i \coth (x)) \coth (x) \, dx\\ &=-\frac {3 x}{2}+\frac {3 \coth (x)}{2}-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+2 \int \coth (x) \, dx\\ &=-\frac {3 x}{2}+\frac {3 \coth (x)}{2}-\coth ^2(x)+\frac {\coth ^3(x)}{2 (1+\coth (x))}+2 \log (\sinh (x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 33, normalized size = 0.89 \begin {gather*} \frac {1}{4} \left (-6 x+\cosh (2 x)+4 \coth (x)-2 \text {csch}^2(x)+8 \log (\sinh (x))-\sinh (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.42, size = 32, normalized size = 0.86
method | result | size |
risch | \(-\frac {7 x}{2}+\frac {{\mathrm e}^{-2 x}}{4}-\frac {2}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+2 \ln \left ({\mathrm e}^{2 x}-1\right )\) | \(30\) |
derivativedivides | \(-\frac {\left (\coth ^{2}\left (x \right )\right )}{2}+\coth \left (x \right )-\frac {\ln \left (\coth \left (x \right )-1\right )}{4}-\frac {1}{2 \left (1+\coth \left (x \right )\right )}-\frac {7 \ln \left (1+\coth \left (x \right )\right )}{4}\) | \(32\) |
default | \(-\frac {\left (\coth ^{2}\left (x \right )\right )}{2}+\coth \left (x \right )-\frac {\ln \left (\coth \left (x \right )-1\right )}{4}-\frac {1}{2 \left (1+\coth \left (x \right )\right )}-\frac {7 \ln \left (1+\coth \left (x \right )\right )}{4}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 54, normalized size = 1.46 \begin {gather*} \frac {1}{2} \, x + \frac {2 \, {\left (2 \, e^{\left (-2 \, x\right )} - 1\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{4} \, e^{\left (-2 \, x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 \, \log \left (e^{\left (-x\right )} - 1\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. 357 vs.
\(2 (31) = 62\).
time = 0.39, size = 357, normalized size = 9.65 \begin {gather*} -\frac {14 \, x \cosh \left (x\right )^{6} + 84 \, x \cosh \left (x\right ) \sinh \left (x\right )^{5} + 14 \, x \sinh \left (x\right )^{6} - {\left (28 \, x + 1\right )} \cosh \left (x\right )^{4} + {\left (210 \, x \cosh \left (x\right )^{2} - 28 \, x - 1\right )} \sinh \left (x\right )^{4} + 4 \, {\left (70 \, x \cosh \left (x\right )^{3} - {\left (28 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (7 \, x + 5\right )} \cosh \left (x\right )^{2} + 2 \, {\left (105 \, x \cosh \left (x\right )^{4} - 3 \, {\left (28 \, x + 1\right )} \cosh \left (x\right )^{2} + 7 \, x + 5\right )} \sinh \left (x\right )^{2} - 8 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} - 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} - 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (21 \, x \cosh \left (x\right )^{5} - {\left (28 \, x + 1\right )} \cosh \left (x\right )^{3} + {\left (7 \, x + 5\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1}{4 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} - 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} - 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\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. 197 vs.
\(2 (34) = 68\).
time = 0.76, size = 197, normalized size = 5.32 \begin {gather*} \frac {x \tanh ^{3}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac {x \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} - \frac {4 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh ^{3}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} - \frac {4 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac {4 \log {\left (\tanh {\left (x \right )} \right )} \tanh ^{3}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac {4 \log {\left (\tanh {\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac {3 \tanh ^{2}{\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} + \frac {\tanh {\left (x \right )}}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} - \frac {1}{2 \tanh ^{3}{\left (x \right )} + 2 \tanh ^{2}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 40, normalized size = 1.08 \begin {gather*} -\frac {7}{2} \, x + \frac {{\left (e^{\left (4 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 29, normalized size = 0.78 \begin {gather*} \frac {x}{2}-2\,\ln \left (\mathrm {coth}\left (x\right )+1\right )+\mathrm {coth}\left (x\right )-\frac {{\mathrm {coth}\left (x\right )}^2}{2}-\frac {1}{2\,\left (\mathrm {coth}\left (x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________