Optimal. Leaf size=12 \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0093475, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4122, 191} \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\csc ^2(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{\sqrt{\csc ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0072935, size = 12, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.127, size = 26, normalized size = 2.2 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{-2+2\,\cos \left ( x \right ) }{\frac{1}{\sqrt{- \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.50187, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{\sqrt{\tan \left (x\right )^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.463214, size = 12, normalized size = 1. \begin{align*} -\cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.47557, size = 12, normalized size = 1. \begin{align*} - \frac{\cot{\left (x \right )}}{\sqrt{\csc ^{2}{\left (x \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.32864, size = 15, normalized size = 1.25 \begin{align*} -\cos \left (x\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) + \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]