Optimal. Leaf size=65 \[ \frac {2}{3} (3 x+2) \sin ^3(x)+4 (3 x+2) \sin (x)-\frac {2}{3} \cos ^3(x)-\frac {2}{3} (3 x+2)^2 \cos (x)+14 \cos (x)-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x) \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3311, 3296, 2638, 2633} \[ \frac {2}{3} (3 x+2) \sin ^3(x)+4 (3 x+2) \sin (x)-\frac {2}{3} \cos ^3(x)-\frac {2}{3} (3 x+2)^2 \cos (x)+14 \cos (x)-\frac {1}{3} (3 x+2)^2 \sin ^2(x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2638
Rule 3296
Rule 3311
Rubi steps
\begin {align*} \int (2+3 x)^2 \sin ^3(x) \, dx &=-\frac {1}{3} (2+3 x)^2 \cos (x) \sin ^2(x)+\frac {2}{3} (2+3 x) \sin ^3(x)+\frac {2}{3} \int (2+3 x)^2 \sin (x) \, dx-2 \int \sin ^3(x) \, dx\\ &=-\frac {2}{3} (2+3 x)^2 \cos (x)-\frac {1}{3} (2+3 x)^2 \cos (x) \sin ^2(x)+\frac {2}{3} (2+3 x) \sin ^3(x)+2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )+4 \int (2+3 x) \cos (x) \, dx\\ &=2 \cos (x)-\frac {2}{3} (2+3 x)^2 \cos (x)-\frac {2 \cos ^3(x)}{3}+4 (2+3 x) \sin (x)-\frac {1}{3} (2+3 x)^2 \cos (x) \sin ^2(x)+\frac {2}{3} (2+3 x) \sin ^3(x)-12 \int \sin (x) \, dx\\ &=14 \cos (x)-\frac {2}{3} (2+3 x)^2 \cos (x)-\frac {2 \cos ^3(x)}{3}+4 (2+3 x) \sin (x)-\frac {1}{3} (2+3 x)^2 \cos (x) \sin ^2(x)+\frac {2}{3} (2+3 x) \sin ^3(x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 50, normalized size = 0.77 \[ \frac {1}{12} \left (-9 \left (9 x^2+12 x-14\right ) \cos (x)+\left (9 x^2+12 x+2\right ) \cos (3 x)-2 (3 x+2) (\sin (3 x)-27 \sin (x))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.65, size = 50, normalized size = 0.77 \[ \frac {1}{3} \, {\left (9 \, x^{2} + 12 \, x + 2\right )} \cos \relax (x)^{3} - {\left (9 \, x^{2} + 12 \, x - 10\right )} \cos \relax (x) - \frac {2}{3} \, {\left ({\left (3 \, x + 2\right )} \cos \relax (x)^{2} - 21 \, x - 14\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.98, size = 51, normalized size = 0.78 \[ \frac {1}{12} \, {\left (9 \, x^{2} + 12 \, x + 2\right )} \cos \left (3 \, x\right ) - \frac {3}{4} \, {\left (9 \, x^{2} + 12 \, x - 14\right )} \cos \relax (x) - \frac {1}{6} \, {\left (3 \, x + 2\right )} \sin \left (3 \, x\right ) + \frac {9}{2} \, {\left (3 \, x + 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 62, normalized size = 0.95 \[ -3 x^{2} \left (2+\sin ^{2}\relax (x )\right ) \cos \relax (x )+12 \cos \relax (x )+12 x \sin \relax (x )+2 \left (\sin ^{3}\relax (x )\right ) x -\frac {2 \left (2+\sin ^{2}\relax (x )\right ) \cos \relax (x )}{3}-4 x \left (2+\sin ^{2}\relax (x )\right ) \cos \relax (x )+\frac {4 \left (\sin ^{3}\relax (x )\right )}{3}+8 \sin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 66, normalized size = 1.02 \[ \frac {4}{3} \, \cos \relax (x)^{3} + \frac {1}{12} \, {\left (9 \, x^{2} - 2\right )} \cos \left (3 \, x\right ) + x \cos \left (3 \, x\right ) - \frac {27}{4} \, {\left (x^{2} - 2\right )} \cos \relax (x) - 9 \, x \cos \relax (x) - \frac {1}{2} \, x \sin \left (3 \, x\right ) + \frac {27}{2} \, x \sin \relax (x) - 4 \, \cos \relax (x) - \frac {1}{3} \, \sin \left (3 \, x\right ) + 9 \, \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 65, normalized size = 1.00 \[ 10\,\cos \relax (x)+\frac {28\,\sin \relax (x)}{3}-9\,x^2\,\cos \relax (x)+4\,x\,{\cos \relax (x)}^3+\frac {2\,{\cos \relax (x)}^3}{3}+3\,x^2\,{\cos \relax (x)}^3-\frac {4\,{\cos \relax (x)}^2\,\sin \relax (x)}{3}-12\,x\,\cos \relax (x)+14\,x\,\sin \relax (x)-2\,x\,{\cos \relax (x)}^2\,\sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 100, normalized size = 1.54 \[ - 9 x^{2} \sin ^{2}{\relax (x )} \cos {\relax (x )} - 6 x^{2} \cos ^{3}{\relax (x )} + 14 x \sin ^{3}{\relax (x )} - 12 x \sin ^{2}{\relax (x )} \cos {\relax (x )} + 12 x \sin {\relax (x )} \cos ^{2}{\relax (x )} - 8 x \cos ^{3}{\relax (x )} + \frac {28 \sin ^{3}{\relax (x )}}{3} + 10 \sin ^{2}{\relax (x )} \cos {\relax (x )} + 8 \sin {\relax (x )} \cos ^{2}{\relax (x )} + \frac {32 \cos ^{3}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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