Optimal. Leaf size=6 \[ x-\cot (x) \]
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Rubi [A] time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3012, 8} \[ x-\cot (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3012
Rubi steps
\begin {align*} \int \csc ^2(x) \left (1+\sin ^2(x)\right ) \, dx &=-\cot (x)+\int 1 \, dx\\ &=x-\cot (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 6, normalized size = 1.00 \[ x-\cot (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 14, normalized size = 2.33 \[ \frac {x \sin \relax (x) - \cos \relax (x)}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 16, normalized size = 2.67 \[ x - \frac {1}{2 \, \tan \left (\frac {1}{2} \, x\right )} + \frac {1}{2} \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 7, normalized size = 1.17 \[ x -\cot \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 8, normalized size = 1.33 \[ x - \frac {1}{\tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.93, size = 6, normalized size = 1.00 \[ x-\mathrm {cot}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.21, size = 3, normalized size = 0.50 \[ x - \cot {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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