Optimal. Leaf size=80 \[ \frac {a \sin (c+d x)}{d^2}-\frac {a x \cos (c+d x)}{d}-\frac {6 b \sin (c+d x)}{d^4}+\frac {6 b x \cos (c+d x)}{d^3}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {b x^3 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3339, 3296, 2637} \[ \frac {a \sin (c+d x)}{d^2}-\frac {a x \cos (c+d x)}{d}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {6 b \sin (c+d x)}{d^4}+\frac {6 b x \cos (c+d x)}{d^3}-\frac {b x^3 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3339
Rubi steps
\begin {align*} \int x \left (a+b x^2\right ) \sin (c+d x) \, dx &=\int \left (a x \sin (c+d x)+b x^3 \sin (c+d x)\right ) \, dx\\ &=a \int x \sin (c+d x) \, dx+b \int x^3 \sin (c+d x) \, dx\\ &=-\frac {a x \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}+\frac {a \int \cos (c+d x) \, dx}{d}+\frac {(3 b) \int x^2 \cos (c+d x) \, dx}{d}\\ &=-\frac {a x \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d^2}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {(6 b) \int x \sin (c+d x) \, dx}{d^2}\\ &=\frac {6 b x \cos (c+d x)}{d^3}-\frac {a x \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d^2}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {(6 b) \int \cos (c+d x) \, dx}{d^3}\\ &=\frac {6 b x \cos (c+d x)}{d^3}-\frac {a x \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}-\frac {6 b \sin (c+d x)}{d^4}+\frac {a \sin (c+d x)}{d^2}+\frac {3 b x^2 \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 57, normalized size = 0.71 \[ \frac {\left (a d^2+3 b \left (d^2 x^2-2\right )\right ) \sin (c+d x)-d x \left (a d^2+b \left (d^2 x^2-6\right )\right ) \cos (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 60, normalized size = 0.75 \[ -\frac {{\left (b d^{3} x^{3} + {\left (a d^{3} - 6 \, b d\right )} x\right )} \cos \left (d x + c\right ) - {\left (3 \, b d^{2} x^{2} + a d^{2} - 6 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 60, normalized size = 0.75 \[ -\frac {{\left (b d^{3} x^{3} + a d^{3} x - 6 \, b d x\right )} \cos \left (d x + c\right )}{d^{4}} + \frac {{\left (3 \, b d^{2} x^{2} + a d^{2} - 6 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 181, normalized size = 2.26 \[ \frac {\frac {b \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}-\frac {3 b c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}+a \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )+\frac {3 b \,c^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}+a c \cos \left (d x +c \right )+\frac {b \,c^{3} \cos \left (d x +c \right )}{d^{2}}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 165, normalized size = 2.06 \[ \frac {a c \cos \left (d x + c\right ) + \frac {b c^{3} \cos \left (d x + c\right )}{d^{2}} - {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a - \frac {3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{2}}{d^{2}} + \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c}{d^{2}} - \frac {{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b}{d^{2}}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 73, normalized size = 0.91 \[ \frac {x\,\cos \left (c+d\,x\right )\,\left (6\,b-a\,d^2\right )}{d^3}-\frac {\sin \left (c+d\,x\right )\,\left (6\,b-a\,d^2\right )}{d^4}-\frac {b\,x^3\,\cos \left (c+d\,x\right )}{d}+\frac {3\,b\,x^2\,\sin \left (c+d\,x\right )}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 99, normalized size = 1.24 \[ \begin {cases} - \frac {a x \cos {\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )}}{d^{2}} - \frac {b x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 b x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 b x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 b \sin {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{2}}{2} + \frac {b x^{4}}{4}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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