Optimal. Leaf size=68 \[ \frac{a (c+d x)^3}{3 d}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2}-\frac{b (c+d x)^2 \cos (e+f x)}{f}+\frac{2 b d^2 \cos (e+f x)}{f^3} \]
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Rubi [A] time = 0.0856717, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3296, 2638} \[ \frac{a (c+d x)^3}{3 d}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2}-\frac{b (c+d x)^2 \cos (e+f x)}{f}+\frac{2 b d^2 \cos (e+f x)}{f^3} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x)^2 (a+b \sin (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \sin (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^3}{3 d}+b \int (c+d x)^2 \sin (e+f x) \, dx\\ &=\frac{a (c+d x)^3}{3 d}-\frac{b (c+d x)^2 \cos (e+f x)}{f}+\frac{(2 b d) \int (c+d x) \cos (e+f x) \, dx}{f}\\ &=\frac{a (c+d x)^3}{3 d}-\frac{b (c+d x)^2 \cos (e+f x)}{f}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2}-\frac{\left (2 b d^2\right ) \int \sin (e+f x) \, dx}{f^2}\\ &=\frac{a (c+d x)^3}{3 d}+\frac{2 b d^2 \cos (e+f x)}{f^3}-\frac{b (c+d x)^2 \cos (e+f x)}{f}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2}\\ \end{align*}
Mathematica [A] time = 0.313986, size = 84, normalized size = 1.24 \[ \frac{1}{3} a x \left (3 c^2+3 c d x+d^2 x^2\right )-\frac{b \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \cos (e+f x)}{f^3}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 241, normalized size = 3.5 \begin{align*}{\frac{1}{f} \left ({\frac{a{d}^{2} \left ( fx+e \right ) ^{3}}{3\,{f}^{2}}}+{\frac{acd \left ( fx+e \right ) ^{2}}{f}}-{\frac{a{d}^{2}e \left ( fx+e \right ) ^{2}}{{f}^{2}}}+a{c}^{2} \left ( fx+e \right ) -2\,{\frac{acde \left ( fx+e \right ) }{f}}+{\frac{a{d}^{2}{e}^{2} \left ( fx+e \right ) }{{f}^{2}}}+{\frac{b{d}^{2} \left ( - \left ( fx+e \right ) ^{2}\cos \left ( fx+e \right ) +2\,\cos \left ( fx+e \right ) +2\, \left ( fx+e \right ) \sin \left ( fx+e \right ) \right ) }{{f}^{2}}}+2\,{\frac{cbd \left ( \sin \left ( fx+e \right ) - \left ( fx+e \right ) \cos \left ( fx+e \right ) \right ) }{f}}-2\,{\frac{b{d}^{2}e \left ( \sin \left ( fx+e \right ) - \left ( fx+e \right ) \cos \left ( fx+e \right ) \right ) }{{f}^{2}}}-{c}^{2}b\cos \left ( fx+e \right ) +2\,{\frac{cbde\cos \left ( fx+e \right ) }{f}}-{\frac{b{d}^{2}{e}^{2}\cos \left ( fx+e \right ) }{{f}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.994145, size = 323, normalized size = 4.75 \begin{align*} \frac{3 \,{\left (f x + e\right )} a c^{2} + \frac{{\left (f x + e\right )}^{3} a d^{2}}{f^{2}} - \frac{3 \,{\left (f x + e\right )}^{2} a d^{2} e}{f^{2}} + \frac{3 \,{\left (f x + e\right )} a d^{2} e^{2}}{f^{2}} + \frac{3 \,{\left (f x + e\right )}^{2} a c d}{f} - \frac{6 \,{\left (f x + e\right )} a c d e}{f} - 3 \, b c^{2} \cos \left (f x + e\right ) - \frac{3 \, b d^{2} e^{2} \cos \left (f x + e\right )}{f^{2}} + \frac{6 \, b c d e \cos \left (f x + e\right )}{f} + \frac{6 \,{\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} b d^{2} e}{f^{2}} - \frac{6 \,{\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} b c d}{f} - \frac{3 \,{\left ({\left ({\left (f x + e\right )}^{2} - 2\right )} \cos \left (f x + e\right ) - 2 \,{\left (f x + e\right )} \sin \left (f x + e\right )\right )} b d^{2}}{f^{2}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68959, size = 228, normalized size = 3.35 \begin{align*} \frac{a d^{2} f^{3} x^{3} + 3 \, a c d f^{3} x^{2} + 3 \, a c^{2} f^{3} x - 3 \,{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} - 2 \, b d^{2}\right )} \cos \left (f x + e\right ) + 6 \,{\left (b d^{2} f x + b c d f\right )} \sin \left (f x + e\right )}{3 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.813716, size = 151, normalized size = 2.22 \begin{align*} \begin{cases} a c^{2} x + a c d x^{2} + \frac{a d^{2} x^{3}}{3} - \frac{b c^{2} \cos{\left (e + f x \right )}}{f} - \frac{2 b c d x \cos{\left (e + f x \right )}}{f} + \frac{2 b c d \sin{\left (e + f x \right )}}{f^{2}} - \frac{b d^{2} x^{2} \cos{\left (e + f x \right )}}{f} + \frac{2 b d^{2} x \sin{\left (e + f x \right )}}{f^{2}} + \frac{2 b d^{2} \cos{\left (e + f x \right )}}{f^{3}} & \text{for}\: f \neq 0 \\\left (a + b \sin{\left (e \right )}\right ) \left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.49884, size = 128, normalized size = 1.88 \begin{align*} \frac{1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x - \frac{{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} - 2 \, b d^{2}\right )} \cos \left (f x + e\right )}{f^{3}} + \frac{2 \,{\left (b d^{2} f x + b c d f\right )} \sin \left (f x + e\right )}{f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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