Optimal. Leaf size=156 \[ -\frac {6 a \sin (c+d x)}{d^4}+\frac {6 a x \cos (c+d x)}{d^3}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {a x^3 \cos (c+d x)}{d}+\frac {720 b \cos (c+d x)}{d^7}+\frac {720 b x \sin (c+d x)}{d^6}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {30 b x^4 \cos (c+d x)}{d^3}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {b x^6 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3339, 3296, 2637, 2638} \[ \frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {6 a \sin (c+d x)}{d^4}+\frac {6 a x \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}+\frac {720 b x \sin (c+d x)}{d^6}+\frac {720 b \cos (c+d x)}{d^7}-\frac {b x^6 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3339
Rubi steps
\begin {align*} \int x^3 \left (a+b x^3\right ) \sin (c+d x) \, dx &=\int \left (a x^3 \sin (c+d x)+b x^6 \sin (c+d x)\right ) \, dx\\ &=a \int x^3 \sin (c+d x) \, dx+b \int x^6 \sin (c+d x) \, dx\\ &=-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^6 \cos (c+d x)}{d}+\frac {(3 a) \int x^2 \cos (c+d x) \, dx}{d}+\frac {(6 b) \int x^5 \cos (c+d x) \, dx}{d}\\ &=-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^6 \cos (c+d x)}{d}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {(6 a) \int x \sin (c+d x) \, dx}{d^2}-\frac {(30 b) \int x^4 \sin (c+d x) \, dx}{d^2}\\ &=\frac {6 a x \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {(6 a) \int \cos (c+d x) \, dx}{d^3}-\frac {(120 b) \int x^3 \cos (c+d x) \, dx}{d^3}\\ &=\frac {6 a x \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}+\frac {(360 b) \int x^2 \sin (c+d x) \, dx}{d^4}\\ &=\frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}+\frac {(720 b) \int x \cos (c+d x) \, dx}{d^5}\\ &=\frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {720 b x \sin (c+d x)}{d^6}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}-\frac {(720 b) \int \sin (c+d x) \, dx}{d^6}\\ &=\frac {720 b \cos (c+d x)}{d^7}+\frac {6 a x \cos (c+d x)}{d^3}-\frac {360 b x^2 \cos (c+d x)}{d^5}-\frac {a x^3 \cos (c+d x)}{d}+\frac {30 b x^4 \cos (c+d x)}{d^3}-\frac {b x^6 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}+\frac {720 b x \sin (c+d x)}{d^6}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {120 b x^3 \sin (c+d x)}{d^4}+\frac {6 b x^5 \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 101, normalized size = 0.65 \[ \frac {3 d \left (a d^2 \left (d^2 x^2-2\right )+2 b x \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \sin (c+d x)-\left (a d^4 x \left (d^2 x^2-6\right )+b \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \cos (c+d x)}{d^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 104, normalized size = 0.67 \[ -\frac {{\left (b d^{6} x^{6} + a d^{6} x^{3} - 30 \, b d^{4} x^{4} - 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 720 \, b\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} - 40 \, b d^{3} x^{3} - 2 \, a d^{3} + 240 \, b d x\right )} \sin \left (d x + c\right )}{d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 106, normalized size = 0.68 \[ -\frac {{\left (b d^{6} x^{6} + a d^{6} x^{3} - 30 \, b d^{4} x^{4} - 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 720 \, b\right )} \cos \left (d x + c\right )}{d^{7}} + \frac {3 \, {\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} - 40 \, b d^{3} x^{3} - 2 \, a d^{3} + 240 \, b d x\right )} \sin \left (d x + c\right )}{d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 556, normalized size = 3.56 \[ \frac {\frac {b \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}+\frac {15 b \,c^{2} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}+a \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 )-\frac {20 b \,c^{3} \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}}-3 a 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 )+\frac {15 b \,c^{4} \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}}+3 a \,c^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )-\frac {6 b \,c^{5} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}+a \,c^{3} \cos \left (d x +c \right )-\frac {b \,c^{6} \cos \left (d x +c \right )}{d^{3}}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 449, normalized size = 2.88 \[ \frac {a c^{3} \cos \left (d x + c\right ) - \frac {b c^{6} \cos \left (d x + c\right )}{d^{3}} - 3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c^{2} + \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{5}}{d^{3}} + 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 )} a c - \frac {15 \, {\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^{4}}{d^{3}} - {\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 )} a + \frac {20 \, {\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 c^{3}}{d^{3}} - \frac {15 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d^{3}} + \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \, {\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b c}{d^{3}} - \frac {{\left ({\left ({\left (d x + c\right )}^{6} - 30 \, {\left (d x + c\right )}^{4} + 360 \, {\left (d x + c\right )}^{2} - 720\right )} \cos \left (d x + c\right ) - 6 \, {\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \sin \left (d x + c\right )\right )} b}{d^{3}}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 151, normalized size = 0.97 \[ \frac {d^4\,\left (6\,a\,x\,\cos \left (c+d\,x\right )+30\,b\,x^4\,\cos \left (c+d\,x\right )\right )+720\,b\,\cos \left (c+d\,x\right )-d^6\,\left (a\,x^3\,\cos \left (c+d\,x\right )+b\,x^6\,\cos \left (c+d\,x\right )\right )+d^5\,\left (3\,a\,x^2\,\sin \left (c+d\,x\right )+6\,b\,x^5\,\sin \left (c+d\,x\right )\right )-d^3\,\left (6\,a\,\sin \left (c+d\,x\right )+120\,b\,x^3\,\sin \left (c+d\,x\right )\right )+720\,b\,d\,x\,\sin \left (c+d\,x\right )-360\,b\,d^2\,x^2\,\cos \left (c+d\,x\right )}{d^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.52, size = 185, normalized size = 1.19 \[ \begin {cases} - \frac {a x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 a x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 a \sin {\left (c + d x \right )}}{d^{4}} - \frac {b x^{6} \cos {\left (c + d x \right )}}{d} + \frac {6 b x^{5} \sin {\left (c + d x \right )}}{d^{2}} + \frac {30 b x^{4} \cos {\left (c + d x \right )}}{d^{3}} - \frac {120 b x^{3} \sin {\left (c + d x \right )}}{d^{4}} - \frac {360 b x^{2} \cos {\left (c + d x \right )}}{d^{5}} + \frac {720 b x \sin {\left (c + d x \right )}}{d^{6}} + \frac {720 b \cos {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{7}}{7}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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