Optimal. Leaf size=17 \[ -\frac {\sin (e+f x) \cos (e+f x)}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4043} \[ -\frac {\sin (e+f x) \cos (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin {align*} \int \cos ^2(e+f x) \left (-2+\sec ^2(e+f x)\right ) \, dx &=-\frac {\cos (e+f x) \sin (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.94 \[ -\frac {\sin (2 e) \cos (2 f x)}{2 f}-\frac {\cos (2 e) \sin (2 f x)}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 17, normalized size = 1.00 \[ -\frac {\cos \left (f x + e\right ) \sin \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 15, normalized size = 0.88 \[ -\frac {\sin \left (2 \, f x + 2 \, e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 18, normalized size = 1.06 \[ -\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 23, normalized size = 1.35 \[ -\frac {\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 14, normalized size = 0.82 \[ -\frac {\sin \left (2\,e+2\,f\,x\right )}{2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.21, size = 49, normalized size = 2.88 \[ x - 2 \left (\begin {cases} \frac {x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {\sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \cos ^{2}{\relax (e )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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