Optimal. Leaf size=118 \[ \frac {2 a^2 (c+d x) \sin (e+f x)}{f}+\frac {a^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {a^2 (c+d x)^2}{2 d}+\frac {1}{2} a^2 c x+\frac {a^2 d \cos ^2(e+f x)}{4 f^2}+\frac {2 a^2 d \cos (e+f x)}{f^2}+\frac {1}{4} a^2 d x^2 \]
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Rubi [A] time = 0.10, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3317, 3296, 2638, 3310} \[ \frac {2 a^2 (c+d x) \sin (e+f x)}{f}+\frac {a^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {a^2 (c+d x)^2}{2 d}+\frac {1}{2} a^2 c x+\frac {a^2 d \cos ^2(e+f x)}{4 f^2}+\frac {2 a^2 d \cos (e+f x)}{f^2}+\frac {1}{4} a^2 d x^2 \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rule 3317
Rubi steps
\begin {align*} \int (c+d x) (a+a \cos (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a^2 (c+d x) \cos (e+f x)+a^2 (c+d x) \cos ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}+a^2 \int (c+d x) \cos ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x) \cos (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}+\frac {a^2 d \cos ^2(e+f x)}{4 f^2}+\frac {2 a^2 (c+d x) \sin (e+f x)}{f}+\frac {a^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} a^2 \int (c+d x) \, dx-\frac {\left (2 a^2 d\right ) \int \sin (e+f x) \, dx}{f}\\ &=\frac {1}{2} a^2 c x+\frac {1}{4} a^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}+\frac {2 a^2 d \cos (e+f x)}{f^2}+\frac {a^2 d \cos ^2(e+f x)}{4 f^2}+\frac {2 a^2 (c+d x) \sin (e+f x)}{f}+\frac {a^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 80, normalized size = 0.68 \[ \frac {a^2 (-6 (e+f x) (d (e-f x)-2 c f)+16 f (c+d x) \sin (e+f x)+2 f (c+d x) \sin (2 (e+f x))+16 d \cos (e+f x)+d \cos (2 (e+f x)))}{8 f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 98, normalized size = 0.83 \[ \frac {3 \, a^{2} d f^{2} x^{2} + 6 \, a^{2} c f^{2} x + a^{2} d \cos \left (f x + e\right )^{2} + 8 \, a^{2} d \cos \left (f x + e\right ) + 2 \, {\left (4 \, a^{2} d f x + 4 \, a^{2} c f + {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 107, normalized size = 0.91 \[ \frac {3}{4} \, a^{2} d x^{2} + \frac {3}{2} \, a^{2} c x + \frac {a^{2} d \cos \left (2 \, f x + 2 \, e\right )}{8 \, f^{2}} + \frac {2 \, a^{2} d \cos \left (f x + e\right )}{f^{2}} + \frac {{\left (a^{2} d f x + a^{2} c f\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f^{2}} + \frac {2 \, {\left (a^{2} d f x + a^{2} c f\right )} \sin \left (f x + e\right )}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 218, normalized size = 1.85 \[ \frac {\frac {a^{2} d \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+a^{2} c \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d e \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {2 a^{2} d \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f}+2 a^{2} c \sin \left (f x +e \right )-\frac {2 a^{2} d e \sin \left (f x +e \right )}{f}+\frac {a^{2} d \left (f x +e \right )^{2}}{2 f}+a^{2} c \left (f x +e \right )-\frac {a^{2} d e \left (f x +e \right )}{f}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 197, normalized size = 1.67 \[ \frac {2 \, {\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c + 8 \, {\left (f x + e\right )} a^{2} c + \frac {4 \, {\left (f x + e\right )}^{2} a^{2} d}{f} - \frac {2 \, {\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d e}{f} - \frac {8 \, {\left (f x + e\right )} a^{2} d e}{f} + 16 \, a^{2} c \sin \left (f x + e\right ) - \frac {16 \, a^{2} d e \sin \left (f x + e\right )}{f} + \frac {{\left (2 \, {\left (f x + e\right )}^{2} + 2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) + \cos \left (2 \, f x + 2 \, e\right )\right )} a^{2} d}{f} + \frac {16 \, {\left ({\left (f x + e\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} a^{2} d}{f}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 117, normalized size = 0.99 \[ \frac {3\,a^2\,d\,f^2\,x^2-16\,a^2\,d\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^2\,d\,{\sin \left (e+f\,x\right )}^2+8\,a^2\,c\,f\,\sin \left (e+f\,x\right )+a^2\,c\,f\,\sin \left (2\,e+2\,f\,x\right )+6\,a^2\,c\,f^2\,x+a^2\,d\,f\,x\,\sin \left (2\,e+2\,f\,x\right )+8\,a^2\,d\,f\,x\,\sin \left (e+f\,x\right )}{4\,f^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 219, normalized size = 1.86 \[ \begin {cases} \frac {a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c x + \frac {a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c \sin {\left (e + f x \right )}}{f} + \frac {a^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac {a^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{2} d x^{2}}{2} + \frac {a^{2} d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} d x \sin {\left (e + f x \right )}}{f} - \frac {a^{2} d \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {2 a^{2} d \cos {\left (e + f x \right )}}{f^{2}} & \text {for}\: f \neq 0 \\\left (a \cos {\relax (e )} + a\right )^{2} \left (c x + \frac {d x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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