Optimal. Leaf size=7 \[ x-\log (\sin (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3556}
\begin {gather*} x-\log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=x-\int \cot (x) \, dx\\ &=x-\log (\sin (x))\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} x-\log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 8, normalized size = 1.14
method | result | size |
default | \(x -\ln \left (\sin \left (x \right )\right )\) | \(8\) |
parts | \(x -\ln \left (\sin \left (x \right )\right )\) | \(8\) |
risch | \(x +i x -\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(17\) |
norman | \(x -\ln \left (\tan \left (\frac {x}{2}\right )\right )+\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )\) | \(19\) |
parallelrisch | \(-\ln \left (-\frac {\cot \left (x \right )}{2}+\frac {\csc \left (x \right )}{2}\right )+\ln \left (\frac {1}{1+\cos \left (x \right )}\right )+x\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 7, normalized size = 1.00 \begin {gather*} x - \log \left (\sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.62, size = 9, normalized size = 1.29 \begin {gather*} x - \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 5, normalized size = 0.71 \begin {gather*} x - \log {\left (\sin {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.47, size = 8, normalized size = 1.14 \begin {gather*} x - \log \left ({\left | \sin \left (x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.17, size = 26, normalized size = 3.71 \begin {gather*} -\ln \left (\mathrm {tan}\left (x\right )\right )+\ln \left (\mathrm {tan}\left (x\right )-\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Chatgpt [F] Failed to verify
time = 1.00, size = 8, normalized size = 1.14 \begin {gather*} \ln \left (\frac {\sin \left (x \right )}{\cos \left (x \right )}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________