Optimal. Leaf size=6 \[ x-\cot (x) \]
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Rubi [A] time = 0.0159719, antiderivative size = 6, normalized size of antiderivative = 1., 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 3012
Rule 8
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*}
Mathematica [A] time = 0.0028742, size = 6, normalized size = 1. \[ x-\cot (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 7, normalized size = 1.2 \begin{align*} x-\cot \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47256, size = 11, normalized size = 1.83 \begin{align*} x - \frac{1}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41083, size = 38, normalized size = 6.33 \begin{align*} \frac{x \sin \left (x\right ) - \cos \left (x\right )}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.1619, size = 3, normalized size = 0.5 \begin{align*} x - \cot{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10361, size = 22, normalized size = 3.67 \begin{align*} x - \frac{1}{2 \, \tan \left (\frac{1}{2} \, x\right )} + \frac{1}{2} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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