Optimal. Leaf size=7 \[ \tan (x)-\cos (x) \]
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Rubi [A] time = 0.0376473, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4401, 3767, 8, 2638} \[ \tan (x)-\cos (x) \]
Antiderivative was successfully verified.
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Rule 4401
Rule 3767
Rule 8
Rule 2638
Rubi steps
\begin{align*} \int \cos (x) \left (\sec ^3(x)+\tan (x)\right ) \, dx &=\int \left (\sec ^2(x)+\sin (x)\right ) \, dx\\ &=\int \sec ^2(x) \, dx+\int \sin (x) \, dx\\ &=-\cos (x)-\operatorname{Subst}(\int 1 \, dx,x,-\tan (x))\\ &=-\cos (x)+\tan (x)\\ \end{align*}
Mathematica [A] time = 0.0036795, size = 7, normalized size = 1. \[ \tan (x)-\cos (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 8, normalized size = 1.1 \begin{align*} -\cos \left ( x \right ) +\tan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959994, size = 9, normalized size = 1.29 \begin{align*} -\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 = 2.36697, size = 39, normalized size = 5.57 \begin{align*} -\frac{\cos \left (x\right )^{2} - \sin \left (x\right )}{\cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.3179, size = 8, normalized size = 1.14 \begin{align*} \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} - \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0972, size = 41, normalized size = 5.86 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{3} + \tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) - 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{4} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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