Optimal. Leaf size=4 \[ -\cot (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3175, 3767, 8} \[ -\cot (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{1-\cos ^2(x)} \, dx &=\int \csc ^2(x) \, dx\\ &=-\operatorname {Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\cot (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 4, normalized size = 1.00 \[ -\cot (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 8, normalized size = 2.00 \[ -\frac {\cos \relax (x)}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 6, normalized size = 1.50 \[ -\frac {1}{\tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 7, normalized size = 1.75 \[ -\frac {1}{\tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 6, normalized size = 1.50 \[ -\frac {1}{\tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.24, size = 4, normalized size = 1.00 \[ -\mathrm {cot}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.43, size = 14, normalized size = 3.50 \[ \frac {\tan {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tan {\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________