Optimal. Leaf size=106 \[ -\frac {2 \left (3 a c d+b \left (c^2+d^2\right )\right ) \cos (e+f x)}{3 f}+\frac {1}{2} x \left (a \left (2 c^2+d^2\right )+2 b c d\right )-\frac {d (3 a d+2 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f} \]
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Rubi [A] time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2734} \[ -\frac {2 \left (3 a c d+b \left (c^2+d^2\right )\right ) \cos (e+f x)}{3 f}+\frac {1}{2} x \left (a \left (2 c^2+d^2\right )+2 b c d\right )-\frac {d (3 a d+2 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rubi steps
\begin {align*} \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx &=-\frac {b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}+\frac {1}{3} \int (c+d \sin (e+f x)) (3 a c+2 b d+(2 b c+3 a d) \sin (e+f x)) \, dx\\ &=\frac {1}{2} \left (2 b c d+a \left (2 c^2+d^2\right )\right ) x-\frac {2 \left (3 a c d+b \left (c^2+d^2\right )\right ) \cos (e+f x)}{3 f}-\frac {d (2 b c+3 a d) \cos (e+f x) \sin (e+f x)}{6 f}-\frac {b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 90, normalized size = 0.85 \[ \frac {6 (e+f x) \left (a \left (2 c^2+d^2\right )+2 b c d\right )-3 \left (8 a c d+4 b c^2+3 b d^2\right ) \cos (e+f x)-3 d (a d+2 b c) \sin (2 (e+f x))+b d^2 \cos (3 (e+f x))}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 90, normalized size = 0.85 \[ \frac {2 \, b d^{2} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, a c^{2} + 2 \, b c d + a d^{2}\right )} f x - 3 \, {\left (2 \, b c d + a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \, {\left (b c^{2} + 2 \, a c d + b d^{2}\right )} \cos \left (f x + e\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 96, normalized size = 0.91 \[ \frac {b d^{2} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {1}{2} \, {\left (2 \, a c^{2} + 2 \, b c d + a d^{2}\right )} x - \frac {{\left (4 \, b c^{2} + 8 \, a c d + 3 \, b d^{2}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (2 \, b c d + a d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 115, normalized size = 1.08 \[ \frac {c^{2} a \left (f x +e \right )-2 a c d \cos \left (f x +e \right )+a \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-b \,c^{2} \cos \left (f x +e \right )+2 b c d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{2} b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 112, normalized size = 1.06 \[ \frac {12 \, {\left (f x + e\right )} a c^{2} + 6 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b c d + 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a d^{2} + 4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b d^{2} - 12 \, b c^{2} \cos \left (f x + e\right ) - 24 \, a c d \cos \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.89, size = 108, normalized size = 1.02 \[ -\frac {\frac {3\,a\,d^2\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {b\,d^2\,\cos \left (3\,e+3\,f\,x\right )}{2}+6\,b\,c^2\,\cos \left (e+f\,x\right )+\frac {9\,b\,d^2\,\cos \left (e+f\,x\right )}{2}+3\,b\,c\,d\,\sin \left (2\,e+2\,f\,x\right )-6\,a\,c^2\,f\,x-3\,a\,d^2\,f\,x+12\,a\,c\,d\,\cos \left (e+f\,x\right )-6\,b\,c\,d\,f\,x}{6\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 199, normalized size = 1.88 \[ \begin {cases} a c^{2} x - \frac {2 a c d \cos {\left (e + f x \right )}}{f} + \frac {a d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {b c^{2} \cos {\left (e + f x \right )}}{f} + b c d x \sin ^{2}{\left (e + f x \right )} + b c d x \cos ^{2}{\left (e + f x \right )} - \frac {b c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {b d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 b d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\relax (e )}\right ) \left (c + d \sin {\relax (e )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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