Optimal. Leaf size=19 \[ -\frac {\sin (e+f x) \cos ^2(e+f x)}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4043} \[ -\frac {\sin (e+f x) \cos ^2(e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin {align*} \int \cos ^3(e+f x) \left (-3+2 \sec ^2(e+f x)\right ) \, dx &=-\frac {\cos ^2(e+f x) \sin (e+f x)}{f}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 46, normalized size = 2.42 \[ \frac {\sin ^3(e+f x)}{f}-\frac {3 \sin (e+f x)}{f}+\frac {2 \sin (e) \cos (f x)}{f}+\frac {2 \cos (e) \sin (f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 19, normalized size = 1.00 \[ -\frac {\cos \left (f x + e\right )^{2} \sin \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 23, normalized size = 1.21 \[ \frac {\sin \left (f x + e\right )^{3} - \sin \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.72, size = 32, normalized size = 1.68 \[ \frac {-\left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+2 \sin \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 21, normalized size = 1.11 \[ \frac {\sin \left (f x + e\right )^{3} - \sin \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 22, normalized size = 1.16 \[ -\frac {\sin \left (e+f\,x\right )-{\sin \left (e+f\,x\right )}^3}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2 \sec ^{2}{\left (e + f x \right )} - 3\right ) \cos ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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