Optimal. Leaf size=68 \[ -\frac {a \cos (c+d x)}{d}-\frac {6 b \sin (c+d x)}{d^4}+\frac {6 b x \cos (c+d x)}{d^3}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {b x^3 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3329, 2638, 3296, 2637} \[ -\frac {a \cos (c+d x)}{d}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {6 b \sin (c+d x)}{d^4}+\frac {6 b x \cos (c+d x)}{d^3}-\frac {b x^3 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3329
Rubi steps
\begin {align*} \int \left (a+b x^3\right ) \sin (c+d x) \, dx &=\int \left (a \sin (c+d x)+b x^3 \sin (c+d x)\right ) \, dx\\ &=a \int \sin (c+d x) \, dx+b \int x^3 \sin (c+d x) \, dx\\ &=-\frac {a \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}+\frac {(3 b) \int x^2 \cos (c+d x) \, dx}{d}\\ &=-\frac {a \cos (c+d x)}{d}-\frac {b x^3 \cos (c+d x)}{d}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {(6 b) \int x \sin (c+d x) \, dx}{d^2}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {6 b x \cos (c+d x)}{d^3}-\frac {b x^3 \cos (c+d x)}{d}+\frac {3 b x^2 \sin (c+d x)}{d^2}-\frac {(6 b) \int \cos (c+d x) \, dx}{d^3}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {6 b x \cos (c+d x)}{d^3}-\frac {b x^3 \cos (c+d x)}{d}-\frac {6 b \sin (c+d x)}{d^4}+\frac {3 b x^2 \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 50, normalized size = 0.74 \[ \frac {3 b \left (d^2 x^2-2\right ) \sin (c+d x)-d \left (a d^2+b x \left (d^2 x^2-6\right )\right ) \cos (c+d x)}{d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 52, normalized size = 0.76 \[ -\frac {{\left (b d^{3} x^{3} + a d^{3} - 6 \, b d x\right )} \cos \left (d x + c\right ) - 3 \, {\left (b d^{2} x^{2} - 2 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 54, normalized size = 0.79 \[ -\frac {{\left (b d^{3} x^{3} + a d^{3} - 6 \, b d x\right )} \cos \left (d x + c\right )}{d^{4}} + \frac {3 \, {\left (b d^{2} x^{2} - 2 \, b\right )} \sin \left (d x + c\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 159, normalized size = 2.34 \[ \frac {\frac {b \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}-\frac {3 b c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+\frac {3 b \,c^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}-a \cos \left (d x +c \right )+\frac {b \,c^{3} \cos \left (d x +c \right )}{d^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.73, size = 141, normalized size = 2.07 \[ -\frac {a \cos \left (d x + c\right ) - \frac {b c^{3} \cos \left (d x + c\right )}{d^{3}} + \frac {3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{2}}{d^{3}} - \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c}{d^{3}} + \frac {{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b}{d^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 65, normalized size = 0.96 \[ -\frac {6\,b\,\sin \left (c+d\,x\right )+d^3\,\left (a\,\cos \left (c+d\,x\right )+b\,x^3\,\cos \left (c+d\,x\right )\right )-3\,b\,d^2\,x^2\,\sin \left (c+d\,x\right )-6\,b\,d\,x\,\cos \left (c+d\,x\right )}{d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 82, normalized size = 1.21 \[ \begin {cases} - \frac {a \cos {\left (c + d x \right )}}{d} - \frac {b x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 b x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 b x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 b \sin {\left (c + d x \right )}}{d^{4}} & \text {for}\: d \neq 0 \\\left (a x + \frac {b x^{4}}{4}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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