Optimal. Leaf size=9 \[ \frac {1}{4} \log ^2(\tanh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3855, 6818}
\begin {gather*} \frac {1}{4} \log ^2(\tanh (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3855
Rule 6818
Rubi steps
\begin {align*} \int \text {csch}(2 x) \log (\tanh (x)) \, dx &=\frac {1}{4} \log ^2(\tanh (x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log ^2(\tanh (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.36, size = 8, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\tanh \left (x \right )\right )^{2}}{4}\) | \(8\) |
default | \(\frac {\ln \left (\tanh \left (x \right )\right )^{2}}{4}\) | \(8\) |
risch | \(\frac {\ln \left (1+{\mathrm e}^{2 x}\right )^{2}}{4}-\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) \ln \left (1+{\mathrm e}^{2 x}\right )}{2}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )^{2}}{4}+\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )}{4}-\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )^{2}}{4}-\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )^{2}}{4}+\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )^{3}}{4}-\frac {i \ln \left ({\mathrm e}^{2 x}-1\right ) \pi \,\mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )}{4}+\frac {i \ln \left ({\mathrm e}^{2 x}-1\right ) \pi \,\mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )^{2}}{4}+\frac {i \ln \left ({\mathrm e}^{2 x}-1\right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )^{2}}{4}-\frac {i \ln \left ({\mathrm e}^{2 x}-1\right ) \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\right )^{3}}{4}\) | \(372\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 7, normalized size = 0.78 \begin {gather*} \frac {1}{4} \, \log \left (\tanh \left (x\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 12, normalized size = 1.33 \begin {gather*} \frac {1}{4} \, \log \left (\frac {\sinh \left (x\right )}{\cosh \left (x\right )}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \log {\left (\tanh {\left (x \right )} \right )} \operatorname {csch}{\left (2 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs.
\(2 (7) = 14\).
time = 0.39, size = 20, normalized size = 2.22 \begin {gather*} \frac {1}{4} \, \log \left (\frac {e^{\left (2 \, x\right )} - 1}{e^{\left (2 \, x\right )} + 1}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.67, size = 21, normalized size = 2.33 \begin {gather*} \frac {{\left (\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\ln \left ({\mathrm {e}}^{2\,x}+1\right )\right )}^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________