Optimal. Leaf size=14 \[ -x-\frac{2 \cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.0631115, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4391, 2670, 2680, 8} \[ -x-\frac{2 \cos (x)}{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int (-\sec (x)+\tan (x))^2 \, dx &=\int \sec ^2(x) (-1+\sin (x))^2 \, dx\\ &=\int \frac{\cos ^2(x)}{(-1-\sin (x))^2} \, dx\\ &=-\frac{2 \cos (x)}{1+\sin (x)}-\int 1 \, dx\\ &=-x-\frac{2 \cos (x)}{1+\sin (x)}\\ \end{align*}
Mathematica [A] time = 0.0067543, size = 12, normalized size = 0.86 \[ -x+2 \tan (x)-2 \sec (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 15, normalized size = 1.1 \begin{align*} 2\,\tan \left ( x \right ) -x-2\, \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 = 1.47501, size = 19, normalized size = 1.36 \begin{align*} -x - \frac{2}{\cos \left (x\right )} + 2 \, \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04946, size = 89, normalized size = 6.36 \begin{align*} -\frac{{\left (x + 2\right )} \cos \left (x\right ) +{\left (x - 2\right )} \sin \left (x\right ) + x + 2}{\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 = 1.11767, size = 10, normalized size = 0.71 \begin{align*} - x + 2 \tan{\left (x \right )} - 2 \sec{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06971, size = 19, normalized size = 1.36 \begin{align*} -x - \frac{4}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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