Optimal. Leaf size=14 \[ -x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8}
\begin {gather*} -x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3554
Rubi steps
\begin {align*} \int \cot ^2\left (\frac {3 x}{4}\right ) \, dx &=-\frac {4}{3} \cot \left (\frac {3 x}{4}\right )-\int 1 \, dx\\ &=-x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 28, normalized size = 2.00 \begin {gather*} -\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2\left (\frac {3 x}{4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 18, normalized size = 1.29
method | result | size |
norman | \(\frac {-\frac {4}{3}-x \tan \left (\frac {3 x}{4}\right )}{\tan \left (\frac {3 x}{4}\right )}\) | \(17\) |
risch | \(-x -\frac {8 i}{3 \left ({\mathrm e}^{\frac {3 i x}{2}}-1\right )}\) | \(17\) |
derivativedivides | \(-\frac {4 \cot \left (\frac {3 x}{4}\right )}{3}+\frac {2 \pi }{3}-\frac {4 \,\mathrm {arccot}\left (\cot \left (\frac {3 x}{4}\right )\right )}{3}\) | \(18\) |
default | \(-\frac {4 \cot \left (\frac {3 x}{4}\right )}{3}+\frac {2 \pi }{3}-\frac {4 \,\mathrm {arccot}\left (\cot \left (\frac {3 x}{4}\right )\right )}{3}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.84, size = 12, normalized size = 0.86 \begin {gather*} -x - \frac {4}{3 \, \tan \left (\frac {3}{4} \, 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. 23 vs.
\(2 (10) = 20\).
time = 0.58, size = 23, normalized size = 1.64 \begin {gather*} -\frac {3 \, x \sin \left (\frac {3}{2} \, x\right ) + 4 \, \cos \left (\frac {3}{2} \, x\right ) + 4}{3 \, \sin \left (\frac {3}{2} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 19, normalized size = 1.36 \begin {gather*} - x - \frac {4 \cos {\left (\frac {3 x}{4} \right )}}{3 \sin {\left (\frac {3 x}{4} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.04, size = 18, normalized size = 1.29 \begin {gather*} -x - \frac {2}{3 \, \tan \left (\frac {3}{8} \, x\right )} + \frac {2}{3} \, \tan \left (\frac {3}{8} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.17, size = 10, normalized size = 0.71 \begin {gather*} -x-\frac {4\,\mathrm {cot}\left (\frac {3\,x}{4}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________