Optimal. Leaf size=18 \[ \frac {\sin ^3(e+f x) \cos (e+f x)}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 18, 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 = {3011} \[ \frac {\sin ^3(e+f x) \cos (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3011
Rubi steps
\begin {align*} \int \sin ^2(e+f x) \left (3-4 \sin ^2(e+f x)\right ) \, dx &=\frac {\cos (e+f x) \sin ^3(e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 31, normalized size = 1.72 \[ \frac {2 \sin (2 (e+f x))-\sin (4 (e+f x))+4 e}{8 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 28, normalized size = 1.56 \[ -\frac {{\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 31, normalized size = 1.72 \[ -\frac {\sin \left (4 \, f x + 4 \, e\right )}{8 \, f} + \frac {\sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 44, normalized size = 2.44 \[ \frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )-\frac {3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 34, normalized size = 1.89 \[ \frac {\tan \left (f x + e\right )^{3}}{{\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.24, size = 18, normalized size = 1.00 \[ \frac {\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^3}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 148, normalized size = 8.22 \[ \begin {cases} - \frac {3 x \sin ^{4}{\left (e + f x \right )}}{2} - 3 x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )} + \frac {3 x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {3 x \cos ^{4}{\left (e + f x \right )}}{2} + \frac {3 x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {5 \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{2 f} - \frac {3 \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (3 - 4 \sin ^{2}{\relax (e )}\right ) \sin ^{2}{\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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