Optimal. Leaf size=90 \[ -\frac {2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}+\frac {1}{2} a x \left (2 a^2+3 b^2\right )-\frac {5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2656, 2734} \[ -\frac {2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}+\frac {1}{2} a x \left (2 a^2+3 b^2\right )-\frac {5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
Antiderivative was successfully verified.
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Rule 2656
Rule 2734
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^3 \, dx &=-\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}+\frac {1}{3} \int (a+b \sin (e+f x)) \left (3 a^2+2 b^2+5 a b \sin (e+f x)\right ) \, dx\\ &=\frac {1}{2} a \left (2 a^2+3 b^2\right ) x-\frac {2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}-\frac {5 a b^2 \cos (e+f x) \sin (e+f x)}{6 f}-\frac {b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 71, normalized size = 0.79 \[ \frac {6 a \left (2 a^2+3 b^2\right ) (e+f x)-9 b \left (4 a^2+b^2\right ) \cos (e+f x)-9 a b^2 \sin (2 (e+f x))+b^3 \cos (3 (e+f x))}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 71, normalized size = 0.79 \[ \frac {2 \, b^{3} \cos \left (f x + e\right )^{3} - 9 \, a b^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} f x - 6 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 75, normalized size = 0.83 \[ \frac {b^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, a b^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{2} \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 76, normalized size = 0.84 \[ \frac {-\frac {b^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a \,b^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{2} b \cos \left (f x +e \right )+\left (f x +e \right ) a^{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 74, normalized size = 0.82 \[ a^{3} x + \frac {3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2}}{4 \, f} + \frac {{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3}}{3 \, f} - \frac {3 \, a^{2} b \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.74, size = 127, normalized size = 1.41 \[ a^3\,x-\frac {4\,b^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{f}+\frac {8\,b^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{3\,f}+\frac {3\,a\,b^2\,x}{2}-\frac {6\,a^2\,b\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{f}-\frac {6\,a\,b^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f}+\frac {3\,a\,b^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 128, normalized size = 1.42 \[ \begin {cases} a^{3} x - \frac {3 a^{2} b \cos {\left (e + f x \right )}}{f} + \frac {3 a b^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a b^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 a b^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {b^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 b^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\relax (e )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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