Optimal. Leaf size=12 \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 12, 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 = {3657, 4122, 191} \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\cot ^2(x)}} \, dx &=\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*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\frac {\cot (x)}{\sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 21, normalized size = 1.75 \[ -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 13, normalized size = 1.08 \[ -\frac {\cot \relax (x )}{\sqrt {1+\cot ^{2}\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 10, normalized size = 0.83 \[ -\frac {1}{\sqrt {\tan \relax (x)^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 12, normalized size = 1.00 \[ -\frac {\sin \left (2\,x\right )}{2\,\sqrt {{\sin \relax (x)}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 14, normalized size = 1.17 \[ - \frac {\cot {\relax (x )}}{\sqrt {\cot ^{2}{\relax (x )} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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