Optimal. Leaf size=5 \[ \log \left (\tan \left (e^x\right )\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 2620, 29} \[ \log \left (\tan \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 2282
Rule 2620
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*}
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Mathematica [B] time = 0.02, size = 21, normalized size = 4.20 \[ 2 \left (\frac {1}{2} \log \left (\sin \left (e^x\right )\right )-\frac {1}{2} \log \left (\cos \left (e^x\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 21, normalized size = 4.20 \[ -\frac {1}{2} \, \log \left (\cos \left (e^{x}\right )^{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{4} \, \cos \left (e^{x}\right )^{2} + \frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 17, normalized size = 3.40 \[ -\frac {1}{2} \, \log \left ({\left | \sin \left (e^{x}\right )^{2} - 1 \right |}\right ) + \log \left ({\left | \sin \left (e^{x}\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 5, normalized size = 1.00 \[ \ln \left (\tan \left ({\mathrm e}^{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 19, normalized size = 3.80 \[ -\frac {1}{2} \, \log \left (\sin \left (e^{x}\right )^{2} - 1\right ) + \frac {1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.51, size = 43, normalized size = 8.60 \[ -\ln \left (-16\,{\mathrm {e}}^{2\,x}-16\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}\right )+\ln \left (16\,{\mathrm {e}}^{2\,x}-16\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x} \csc {\left (e^{x} \right )} \sec {\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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