Optimal. Leaf size=50 \[ \frac{2 d (c+d x) \sin (a+b x)}{b^2}+\frac{2 d^2 \cos (a+b x)}{b^3}-\frac{(c+d x)^2 \cos (a+b x)}{b} \]
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Rubi [A] time = 0.0389583, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2638} \[ \frac{2 d (c+d x) \sin (a+b x)}{b^2}+\frac{2 d^2 \cos (a+b x)}{b^3}-\frac{(c+d x)^2 \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x)^2 \sin (a+b x) \, dx &=-\frac{(c+d x)^2 \cos (a+b x)}{b}+\frac{(2 d) \int (c+d x) \cos (a+b x) \, dx}{b}\\ &=-\frac{(c+d x)^2 \cos (a+b x)}{b}+\frac{2 d (c+d x) \sin (a+b x)}{b^2}-\frac{\left (2 d^2\right ) \int \sin (a+b x) \, dx}{b^2}\\ &=\frac{2 d^2 \cos (a+b x)}{b^3}-\frac{(c+d x)^2 \cos (a+b x)}{b}+\frac{2 d (c+d x) \sin (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.170771, size = 45, normalized size = 0.9 \[ \frac{2 b d (c+d x) \sin (a+b x)-\cos (a+b x) \left (b^2 (c+d x)^2-2 d^2\right )}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 148, normalized size = 3. \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2} \left ( - \left ( bx+a \right ) ^{2}\cos \left ( bx+a \right ) +2\,\cos \left ( bx+a \right ) +2\, \left ( bx+a \right ) \sin \left ( bx+a \right ) \right ) }{{b}^{2}}}-2\,{\frac{a{d}^{2} \left ( \sin \left ( bx+a \right ) - \left ( bx+a \right ) \cos \left ( bx+a \right ) \right ) }{{b}^{2}}}+2\,{\frac{cd \left ( \sin \left ( bx+a \right ) - \left ( bx+a \right ) \cos \left ( bx+a \right ) \right ) }{b}}-{\frac{{a}^{2}{d}^{2}\cos \left ( bx+a \right ) }{{b}^{2}}}+2\,{\frac{acd\cos \left ( bx+a \right ) }{b}}-{c}^{2}\cos \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03117, size = 190, normalized size = 3.8 \begin{align*} -\frac{c^{2} \cos \left (b x + a\right ) - \frac{2 \, a c d \cos \left (b x + a\right )}{b} + \frac{a^{2} d^{2} \cos \left (b x + a\right )}{b^{2}} + \frac{2 \,{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} c d}{b} - \frac{2 \,{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a d^{2}}{b^{2}} + \frac{{\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \,{\left (b x + a\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6819, size = 138, normalized size = 2.76 \begin{align*} -\frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right ) - 2 \,{\left (b d^{2} x + b c d\right )} \sin \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.636426, size = 112, normalized size = 2.24 \begin{align*} \begin{cases} - \frac{c^{2} \cos{\left (a + b x \right )}}{b} - \frac{2 c d x \cos{\left (a + b x \right )}}{b} - \frac{d^{2} x^{2} \cos{\left (a + b x \right )}}{b} + \frac{2 c d \sin{\left (a + b x \right )}}{b^{2}} + \frac{2 d^{2} x \sin{\left (a + b x \right )}}{b^{2}} + \frac{2 d^{2} \cos{\left (a + b x \right )}}{b^{3}} & \text{for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) \sin{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15278, size = 88, normalized size = 1.76 \begin{align*} -\frac{{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )}{b^{3}} + \frac{2 \,{\left (b d^{2} x + b c d\right )} \sin \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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