Optimal. Leaf size=14 \[ x-\frac {1}{3} \tanh ^3(x)-\tanh (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ x-\frac {1}{3} \tanh ^3(x)-\tanh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \tanh ^4(x) \, dx &=-\frac {1}{3} \tanh ^3(x)+\int \tanh ^2(x) \, dx\\ &=-\tanh (x)-\frac {\tanh ^3(x)}{3}+\int 1 \, dx\\ &=x-\tanh (x)-\frac {\tanh ^3(x)}{3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 18, normalized size = 1.29 \[ x-\frac {4 \tanh (x)}{3}+\frac {1}{3} \tanh (x) \text {sech}^2(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh ^4(x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.28, size = 68, normalized size = 4.86 \[ \frac {{\left (3 \, x + 4\right )} \cosh \relax (x)^{3} + 3 \, {\left (3 \, x + 4\right )} \cosh \relax (x) \sinh \relax (x)^{2} - 12 \, \cosh \relax (x)^{2} \sinh \relax (x) - 4 \, \sinh \relax (x)^{3} + 3 \, {\left (3 \, x + 4\right )} \cosh \relax (x)}{3 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.63, size = 26, normalized size = 1.86 \[ x + \frac {4 \, {\left (3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 2\right )}}{3 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 26, normalized size = 1.86
method | result | size |
derivativedivides | \(-\frac {\left (\tanh ^{3}\relax (x )\right )}{3}-\tanh \relax (x )-\frac {\ln \left (\tanh \relax (x )-1\right )}{2}+\frac {\ln \left (1+\tanh \relax (x )\right )}{2}\) | \(26\) |
default | \(-\frac {\left (\tanh ^{3}\relax (x )\right )}{3}-\tanh \relax (x )-\frac {\ln \left (\tanh \relax (x )-1\right )}{2}+\frac {\ln \left (1+\tanh \relax (x )\right )}{2}\) | \(26\) |
risch | \(x +\frac {4 \,{\mathrm e}^{4 x}+4 \,{\mathrm e}^{2 x}+\frac {8}{3}}{\left (1+{\mathrm e}^{2 x}\right )^{3}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 38, normalized size = 2.71 \[ x - \frac {4 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 12, normalized size = 0.86 \[ -\frac {{\mathrm {tanh}\relax (x)}^3}{3}-\mathrm {tanh}\relax (x)+x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 10, normalized size = 0.71 \[ x - \frac {\tanh ^{3}{\relax (x )}}{3} - \tanh {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________