Optimal. Leaf size=9 \[ -\cot (x)-\csc (x) \]
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Rubi [A] time = 0.0286004, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2669, 3767, 8} \[ -\cot (x)-\csc (x) \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (1+\cos (x)) \csc ^2(x) \, dx &=-\csc (x)+\int \csc ^2(x) \, dx\\ &=-\csc (x)-\operatorname{Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\cot (x)-\csc (x)\\ \end{align*}
Mathematica [A] time = 0.003734, size = 9, normalized size = 1. \[ -\cot (x)-\csc (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 12, normalized size = 1.3 \begin{align*} - \left ( \sin \left ( x \right ) \right ) ^{-1}-\cot \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968491, size = 18, normalized size = 2. \begin{align*} -\frac{1}{\sin \left (x\right )} - \frac{1}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90474, size = 30, normalized size = 3.33 \begin{align*} -\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.76437, size = 8, normalized size = 0.89 \begin{align*} - \cot{\left (x \right )} - \frac{1}{\sin{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09225, size = 11, normalized size = 1.22 \begin{align*} -\frac{1}{\tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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