Optimal. Leaf size=17 \[ -\frac {\cos (e+f x) \sin (e+f x)}{f} \]
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Rubi [A]
time = 0.01, 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 = {4128}
\begin {gather*} -\frac {\sin (e+f x) \cos (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 4128
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 \begin {gather*} -\frac {\cos (2 f x) \sin (2 e)}{2 f}-\frac {\cos (2 e) \sin (2 f x)}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 18, normalized size = 1.06
method | result | size |
risch | \(-\frac {\sin \left (2 f x +2 e \right )}{2 f}\) | \(15\) |
derivativedivides | \(-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{f}\) | \(18\) |
default | \(-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{f}\) | \(18\) |
norman | \(\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 25, normalized size = 1.47 \begin {gather*} -\frac {\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.59, size = 19, normalized size = 1.12 \begin {gather*} -\frac {\cos \left (f x + e\right ) \sin \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.22, size = 49, normalized size = 2.88 \begin {gather*} 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}{\left (e \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 14, normalized size = 0.82 \begin {gather*} -\frac {\sin \left (2 \, f x + 2 \, e\right )}{2 \, f} \end {gather*}
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 \begin {gather*} -\frac {\sin \left (2\,e+2\,f\,x\right )}{2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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