Optimal. Leaf size=6 \[ -\cot \left (e^x\right ) \]
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Rubi [A] time = 0.0174201, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 3767, 8} \[ -\cot \left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int e^x \csc ^2\left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \csc ^2(x) \, dx,x,e^x\right )\\ &=-\operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (e^x\right )\right )\\ &=-\cot \left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0167705, size = 6, normalized size = 1. \[ -\cot \left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 6, normalized size = 1. \begin{align*} -\cot \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03101, size = 9, normalized size = 1.5 \begin{align*} -\frac{1}{\tan \left (e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.439292, size = 27, normalized size = 4.5 \begin{align*} -\frac{\cos \left (e^{x}\right )}{\sin \left (e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.2511, size = 5, normalized size = 0.83 \begin{align*} - \cot{\left (e^{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27695, size = 9, normalized size = 1.5 \begin{align*} -\frac{1}{\tan \left (e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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