Optimal. Leaf size=53 \[ -\frac {a \cos (c+d x)}{d}+\frac {2 b \cos (c+d x)}{d^3}+\frac {2 b x \sin (c+d x)}{d^2}-\frac {b x^2 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3329, 2638, 3296} \[ -\frac {a \cos (c+d x)}{d}+\frac {2 b x \sin (c+d x)}{d^2}+\frac {2 b \cos (c+d x)}{d^3}-\frac {b x^2 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3329
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \sin (c+d x) \, dx &=\int \left (a \sin (c+d x)+b x^2 \sin (c+d x)\right ) \, dx\\ &=a \int \sin (c+d x) \, dx+b \int x^2 \sin (c+d x) \, dx\\ &=-\frac {a \cos (c+d x)}{d}-\frac {b x^2 \cos (c+d x)}{d}+\frac {(2 b) \int x \cos (c+d x) \, dx}{d}\\ &=-\frac {a \cos (c+d x)}{d}-\frac {b x^2 \cos (c+d x)}{d}+\frac {2 b x \sin (c+d x)}{d^2}-\frac {(2 b) \int \sin (c+d x) \, dx}{d^2}\\ &=\frac {2 b \cos (c+d x)}{d^3}-\frac {a \cos (c+d x)}{d}-\frac {b x^2 \cos (c+d x)}{d}+\frac {2 b x \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 41, normalized size = 0.77 \[ \frac {2 b d x \sin (c+d x)-\left (a d^2+b \left (d^2 x^2-2\right )\right ) \cos (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 41, normalized size = 0.77 \[ \frac {2 \, b d x \sin \left (d x + c\right ) - {\left (b d^{2} x^{2} + a d^{2} - 2 \, b\right )} \cos \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 42, normalized size = 0.79 \[ \frac {2 \, b x \sin \left (d x + c\right )}{d^{2}} - \frac {{\left (b d^{2} x^{2} + a d^{2} - 2 \, b\right )} \cos \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 1.87 \[ \frac {\frac {b \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}-\frac {2 b c \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}-a \cos \left (d x +c \right )-\frac {b \,c^{2} \cos \left (d x +c \right )}{d^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 91, normalized size = 1.72 \[ -\frac {a \cos \left (d x + c\right ) + \frac {b c^{2} \cos \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c}{d^{2}} + \frac {{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b}{d^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 49, normalized size = 0.92 \[ \frac {\cos \left (c+d\,x\right )\,\left (2\,b-a\,d^2\right )}{d^3}+\frac {2\,b\,x\,\sin \left (c+d\,x\right )}{d^2}-\frac {b\,x^2\,\cos \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 65, normalized size = 1.23 \[ \begin {cases} - \frac {a \cos {\left (c + d x \right )}}{d} - \frac {b x^{2} \cos {\left (c + d x \right )}}{d} + \frac {2 b x \sin {\left (c + d x \right )}}{d^{2}} + \frac {2 b \cos {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\left (a x + \frac {b x^{3}}{3}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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