Optimal. Leaf size=5 \[ \tan (x)+\sec (x) \]
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Rubi [A] time = 0.0231152, antiderivative size = 5, 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} \[ \tan (x)+\sec (x) \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(x) (1+\sin (x)) \, dx &=\sec (x)+\int \sec ^2(x) \, dx\\ &=\sec (x)-\operatorname{Subst}(\int 1 \, dx,x,-\tan (x))\\ &=\sec (x)+\tan (x)\\ \end{align*}
Mathematica [A] time = 0.0037442, size = 5, normalized size = 1. \[ \tan (x)+\sec (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 8, normalized size = 1.6 \begin{align*} \tan \left ( x \right ) + \left ( \cos \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94676, size = 9, normalized size = 1.8 \begin{align*} \frac{1}{\cos \left (x\right )} + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04508, size = 61, normalized size = 12.2 \begin{align*} \frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.50021, size = 7, normalized size = 1.4 \begin{align*} \tan{\left (x \right )} + \frac{1}{\cos{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08856, size = 14, normalized size = 2.8 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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