Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 3770} \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 3770
Rubi steps
\begin {align*} \int e^x \sec \left (e^x\right ) \, dx &=\operatorname {Subst}\left (\int \sec (x) \, dx,x,e^x\right )\\ &=\tanh ^{-1}\left (\sin \left (e^x\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 5, normalized size = 1.00 \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 19, normalized size = 3.80 \[ \frac {1}{2} \, \log \left (\sin \left (e^{x}\right ) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \left (e^{x}\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.12, size = 29, normalized size = 5.80 \[ \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) + 2 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 9, normalized size = 1.80 \[ \ln \left (\sec \left ({\mathrm e}^{x}\right )+\tan \left ({\mathrm e}^{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 8, normalized size = 1.60 \[ \log \left (\sec \left (e^{x}\right ) + \tan \left (e^{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.79, size = 10, normalized size = 2.00 \[ -\mathrm {atan}\left ({\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.02, size = 10, normalized size = 2.00 \[ \log {\left (\tan {\left (e^{x} \right )} + \sec {\left (e^{x} \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________