Optimal. Leaf size=50 \[ \frac {2 d^2 \cos (a+b x)}{b^3}+\frac {2 d (c+d x) \sin (a+b x)}{b^2}-\frac {(c+d x)^2 \cos (a+b x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 2638
Rule 3296
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*}
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Mathematica [A] time = 0.19, size = 45, normalized size = 0.90 \[ \frac {2 b d (c+d x) \sin (a+b x)-\cos (a+b x) \left (b^2 (c+d x)^2-2 d^2\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 63, normalized size = 1.26 \[ -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \sin \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 65, normalized size = 1.30 \[ -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )}{b^{3}} + \frac {2 \, {\left (b d^{2} x + b c d\right )} \sin \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 148, normalized size = 2.96 \[ \frac {\frac {d^{2} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}-\frac {2 a \,d^{2} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}+\frac {2 c d \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b}-\frac {a^{2} d^{2} \cos \left (b x +a \right )}{b^{2}}+\frac {2 a c d \cos \left (b x +a \right )}{b}-c^{2} \cos \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 141, normalized size = 2.82 \[ -\frac {c^{2} \cos \left (b x + a\right ) - \frac {2 \, a c d \cos \left (b x + a\right )}{b} + \frac {a^{2} d^{2} \cos \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} c d}{b} - \frac {2 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 84, normalized size = 1.68 \[ \frac {\cos \left (a+b\,x\right )\,\left (2\,d^2-b^2\,c^2\right )}{b^3}-\frac {d^2\,x^2\,\cos \left (a+b\,x\right )}{b}+\frac {2\,c\,d\,\sin \left (a+b\,x\right )}{b^2}+\frac {2\,d^2\,x\,\sin \left (a+b\,x\right )}{b^2}-\frac {2\,c\,d\,x\,\cos \left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.73, size = 112, normalized size = 2.24 \[ \begin {cases} - \frac {c^{2} \cos {\left (a + b x \right )}}{b} - \frac {2 c d x \cos {\left (a + b x \right )}}{b} - \frac {d^{2} x^{2} \cos {\left (a + b x \right )}}{b} + \frac {2 c d \sin {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} x \sin {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} \cos {\left (a + b x \right )}}{b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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