Optimal. Leaf size=5 \[ \log \left (\tan \left (e^x\right )\right ) \]
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Rubi [A] time = 0.0215424, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 2620, 29} \[ \log \left (\tan \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2620
Rule 29
Rubi steps
\begin{align*} \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \csc (x) \sec (x) \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\tan \left (e^x\right )\right )\\ &=\log \left (\tan \left (e^x\right )\right )\\ \end{align*}
Mathematica [B] time = 0.0177249, size = 21, normalized size = 4.2 \[ 2 \left (\frac{1}{2} \log \left (\sin \left (e^x\right )\right )-\frac{1}{2} \log \left (\cos \left (e^x\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 5, normalized size = 1. \begin{align*} \ln \left ( \tan \left ({{\rm e}^{x}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07253, size = 26, normalized size = 5.2 \begin{align*} -\frac{1}{2} \, \log \left (\sin \left (e^{x}\right )^{2} - 1\right ) + \frac{1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.475521, size = 74, normalized size = 14.8 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (e^{x}\right )^{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{4} \, \cos \left (e^{x}\right )^{2} + \frac{1}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \csc{\left (e^{x} \right )} \sec{\left (e^{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11425, size = 27, normalized size = 5.4 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) - \frac{1}{2} \, \log \left ({\left | \sin \left (e^{x}\right )^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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