Optimal. Leaf size=14 \[ -x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \]
________________________________________________________________________________________
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 = {3473, 8} \[ -x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
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] time = 0.01, size = 28, normalized size = 2.00 \[ -\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2\left (\frac {3 x}{4}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^2\left (\frac {3 x}{4}\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.00, size = 23, normalized size = 1.64 \[ -\frac {3 \, x \sin \left (\frac {3}{2} \, x\right ) + 4 \, \cos \left (\frac {3}{2} \, x\right ) + 4}{3 \, \sin \left (\frac {3}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.19, size = 18, normalized size = 1.29 \[ -x - \frac {2}{3 \, \tan \left (\frac {3}{8} \, x\right )} + \frac {2}{3} \, \tan \left (\frac {3}{8} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 17, normalized size = 1.21
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 = 0.98, size = 12, normalized size = 0.86 \[ -x - \frac {4}{3 \, \tan \left (\frac {3}{4} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.17, size = 10, normalized size = 0.71 \[ -x-\frac {4\,\mathrm {cot}\left (\frac {3\,x}{4}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.07, size = 19, normalized size = 1.36 \[ - x - \frac {4 \cos {\left (\frac {3 x}{4} \right )}}{3 \sin {\left (\frac {3 x}{4} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________