Optimal. Leaf size=50 \[ \frac {2 b (a+b x) \sin (c+d x)}{d^2}-\frac {(a+b x)^2 \cos (c+d x)}{d}+\frac {2 b^2 \cos (c+d x)}{d^3} \]
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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} \[ \frac {2 b (a+b x) \sin (c+d x)}{d^2}-\frac {(a+b x)^2 \cos (c+d x)}{d}+\frac {2 b^2 \cos (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rubi steps
\begin {align*} \int (a+b x)^2 \sin (c+d x) \, dx &=-\frac {(a+b x)^2 \cos (c+d x)}{d}+\frac {(2 b) \int (a+b x) \cos (c+d x) \, dx}{d}\\ &=-\frac {(a+b x)^2 \cos (c+d x)}{d}+\frac {2 b (a+b x) \sin (c+d x)}{d^2}-\frac {\left (2 b^2\right ) \int \sin (c+d x) \, dx}{d^2}\\ &=\frac {2 b^2 \cos (c+d x)}{d^3}-\frac {(a+b x)^2 \cos (c+d x)}{d}+\frac {2 b (a+b x) \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 57, normalized size = 1.14 \[ \frac {2 b d (a+b x) \sin (c+d x)-\left (a^2 d^2+2 a b d^2 x+b^2 \left (d^2 x^2-2\right )\right ) \cos (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 63, normalized size = 1.26 \[ -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) - 2 \, {\left (b^{2} d x + a b d\right )} \sin \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 65, normalized size = 1.30 \[ -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{3}} + \frac {2 \, {\left (b^{2} d x + a b d\right )} \sin \left (d x + c\right )}{d^{3}} \]
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 \[ \frac {\frac {b^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+\frac {2 a b \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d}-\frac {2 b^{2} c \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}-a^{2} \cos \left (d x +c \right )+\frac {2 a b c \cos \left (d x +c \right )}{d}-\frac {b^{2} c^{2} \cos \left (d x +c \right )}{d^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 141, normalized size = 2.82 \[ -\frac {a^{2} \cos \left (d x + c\right ) + \frac {b^{2} c^{2} \cos \left (d x + c\right )}{d^{2}} - \frac {2 \, a b c \cos \left (d x + c\right )}{d} - \frac {2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c}{d^{2}} + \frac {2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b}{d} + \frac {{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b^{2}}{d^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.70, size = 84, normalized size = 1.68 \[ \frac {\cos \left (c+d\,x\right )\,\left (2\,b^2-a^2\,d^2\right )}{d^3}-\frac {b^2\,x^2\,\cos \left (c+d\,x\right )}{d}+\frac {2\,a\,b\,\sin \left (c+d\,x\right )}{d^2}+\frac {2\,b^2\,x\,\sin \left (c+d\,x\right )}{d^2}-\frac {2\,a\,b\,x\,\cos \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 112, normalized size = 2.24 \[ \begin {cases} - \frac {a^{2} \cos {\left (c + d x \right )}}{d} - \frac {2 a b x \cos {\left (c + d x \right )}}{d} + \frac {2 a b \sin {\left (c + d x \right )}}{d^{2}} - \frac {b^{2} x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 b^{2} x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 b^{2} \cos {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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