Optimal. Leaf size=235 \[ \frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{48 a b x \sin (c+d x)}{d^4}-\frac{48 a b \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{5040 b^2 \sin (c+d x)}{d^8}+\frac{5040 b^2 x \cos (c+d x)}{d^7}-\frac{b^2 x^7 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.326037, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3339, 3296, 2637, 2638} \[ \frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{48 a b x \sin (c+d x)}{d^4}-\frac{48 a b \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{5040 b^2 \sin (c+d x)}{d^8}+\frac{5040 b^2 x \cos (c+d x)}{d^7}-\frac{b^2 x^7 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3296
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int x \left (a+b x^3\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 x \sin (c+d x)+2 a b x^4 \sin (c+d x)+b^2 x^7 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x \sin (c+d x) \, dx+(2 a b) \int x^4 \sin (c+d x) \, dx+b^2 \int x^7 \sin (c+d x) \, dx\\ &=-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \int \cos (c+d x) \, dx}{d}+\frac{(8 a b) \int x^3 \cos (c+d x) \, dx}{d}+\frac{\left (7 b^2\right ) \int x^6 \cos (c+d x) \, dx}{d}\\ &=-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{(24 a b) \int x^2 \sin (c+d x) \, dx}{d^2}-\frac{\left (42 b^2\right ) \int x^5 \sin (c+d x) \, dx}{d^2}\\ &=-\frac{a^2 x \cos (c+d x)}{d}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{(48 a b) \int x \cos (c+d x) \, dx}{d^3}-\frac{\left (210 b^2\right ) \int x^4 \cos (c+d x) \, dx}{d^3}\\ &=-\frac{a^2 x \cos (c+d x)}{d}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}+\frac{(48 a b) \int \sin (c+d x) \, dx}{d^4}+\frac{\left (840 b^2\right ) \int x^3 \sin (c+d x) \, dx}{d^4}\\ &=-\frac{48 a b \cos (c+d x)}{d^5}-\frac{a^2 x \cos (c+d x)}{d}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}+\frac{\left (2520 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d^5}\\ &=-\frac{48 a b \cos (c+d x)}{d^5}-\frac{a^2 x \cos (c+d x)}{d}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{\left (5040 b^2\right ) \int x \sin (c+d x) \, dx}{d^6}\\ &=-\frac{48 a b \cos (c+d x)}{d^5}+\frac{5040 b^2 x \cos (c+d x)}{d^7}-\frac{a^2 x \cos (c+d x)}{d}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{b^2 x^7 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{\left (5040 b^2\right ) \int \cos (c+d x) \, dx}{d^7}\\ &=-\frac{48 a b \cos (c+d x)}{d^5}+\frac{5040 b^2 x \cos (c+d x)}{d^7}-\frac{a^2 x \cos (c+d x)}{d}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}-\frac{b^2 x^7 \cos (c+d x)}{d}-\frac{5040 b^2 \sin (c+d x)}{d^8}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.384909, size = 139, normalized size = 0.59 \[ \frac{\left (a^2 d^6+8 a b d^4 x \left (d^2 x^2-6\right )+7 b^2 \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \sin (c+d x)-d \left (a^2 d^6 x+2 a b d^2 \left (d^4 x^4-12 d^2 x^2+24\right )+b^2 x \left (d^6 x^6-42 d^4 x^4+840 d^2 x^2-5040\right )\right ) \cos (c+d x)}{d^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 822, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15923, size = 894, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64677, size = 355, normalized size = 1.51 \begin{align*} -\frac{{\left (b^{2} d^{7} x^{7} + 2 \, a b d^{7} x^{4} - 42 \, b^{2} d^{5} x^{5} - 24 \, a b d^{5} x^{2} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{3} +{\left (a^{2} d^{7} - 5040 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) -{\left (7 \, b^{2} d^{6} x^{6} + 8 \, a b d^{6} x^{3} - 210 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} - 5040 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.829, size = 284, normalized size = 1.21 \begin{align*} \begin{cases} - \frac{a^{2} x \cos{\left (c + d x \right )}}{d} + \frac{a^{2} \sin{\left (c + d x \right )}}{d^{2}} - \frac{2 a b x^{4} \cos{\left (c + d x \right )}}{d} + \frac{8 a b x^{3} \sin{\left (c + d x \right )}}{d^{2}} + \frac{24 a b x^{2} \cos{\left (c + d x \right )}}{d^{3}} - \frac{48 a b x \sin{\left (c + d x \right )}}{d^{4}} - \frac{48 a b \cos{\left (c + d x \right )}}{d^{5}} - \frac{b^{2} x^{7} \cos{\left (c + d x \right )}}{d} + \frac{7 b^{2} x^{6} \sin{\left (c + d x \right )}}{d^{2}} + \frac{42 b^{2} x^{5} \cos{\left (c + d x \right )}}{d^{3}} - \frac{210 b^{2} x^{4} \sin{\left (c + d x \right )}}{d^{4}} - \frac{840 b^{2} x^{3} \cos{\left (c + d x \right )}}{d^{5}} + \frac{2520 b^{2} x^{2} \sin{\left (c + d x \right )}}{d^{6}} + \frac{5040 b^{2} x \cos{\left (c + d x \right )}}{d^{7}} - \frac{5040 b^{2} \sin{\left (c + d x \right )}}{d^{8}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{2}}{2} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{8}}{8}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16164, size = 217, normalized size = 0.92 \begin{align*} -\frac{{\left (b^{2} d^{7} x^{7} + 2 \, a b d^{7} x^{4} - 42 \, b^{2} d^{5} x^{5} + a^{2} d^{7} x - 24 \, a b d^{5} x^{2} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{3} - 5040 \, b^{2} d x\right )} \cos \left (d x + c\right )}{d^{8}} + \frac{{\left (7 \, b^{2} d^{6} x^{6} + 8 \, a b d^{6} x^{3} - 210 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} - 5040 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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