Optimal. Leaf size=99 \[ -\frac{3 x^2}{4}+\frac{1}{16} (2 x+1)^4-\frac{3 x}{4}+\frac{3}{8} (2 x+1)^2 \sin ^2(2 x+1)-\frac{3}{16} \sin ^2(2 x+1)-\frac{1}{4} (2 x+1)^3 \sin (2 x+1) \cos (2 x+1)+\frac{3}{8} (2 x+1) \sin (2 x+1) \cos (2 x+1) \]
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Rubi [A] time = 0.0595834, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3311, 32, 3310} \[ -\frac{3 x^2}{4}+\frac{1}{16} (2 x+1)^4-\frac{3 x}{4}+\frac{3}{8} (2 x+1)^2 \sin ^2(2 x+1)-\frac{3}{16} \sin ^2(2 x+1)-\frac{1}{4} (2 x+1)^3 \sin (2 x+1) \cos (2 x+1)+\frac{3}{8} (2 x+1) \sin (2 x+1) \cos (2 x+1) \]
Antiderivative was successfully verified.
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Rule 3311
Rule 32
Rule 3310
Rubi steps
\begin{align*} \int (1+2 x)^3 \sin ^2(1+2 x) \, dx &=-\frac{1}{4} (1+2 x)^3 \cos (1+2 x) \sin (1+2 x)+\frac{3}{8} (1+2 x)^2 \sin ^2(1+2 x)+\frac{1}{2} \int (1+2 x)^3 \, dx-\frac{3}{2} \int (1+2 x) \sin ^2(1+2 x) \, dx\\ &=\frac{1}{16} (1+2 x)^4+\frac{3}{8} (1+2 x) \cos (1+2 x) \sin (1+2 x)-\frac{1}{4} (1+2 x)^3 \cos (1+2 x) \sin (1+2 x)-\frac{3}{16} \sin ^2(1+2 x)+\frac{3}{8} (1+2 x)^2 \sin ^2(1+2 x)-\frac{3}{4} \int (1+2 x) \, dx\\ &=-\frac{3 x}{4}-\frac{3 x^2}{4}+\frac{1}{16} (1+2 x)^4+\frac{3}{8} (1+2 x) \cos (1+2 x) \sin (1+2 x)-\frac{1}{4} (1+2 x)^3 \cos (1+2 x) \sin (1+2 x)-\frac{3}{16} \sin ^2(1+2 x)+\frac{3}{8} (1+2 x)^2 \sin ^2(1+2 x)\\ \end{align*}
Mathematica [A] time = 0.229936, size = 55, normalized size = 0.56 \[ \frac{1}{32} \left (2 (2 x+1) \left (\left (-8 x^2-8 x+1\right ) \sin (4 x+2)+(2 x+1)^3\right )-3 \left (8 x^2+8 x+1\right ) \cos (4 x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 97, normalized size = 1. \begin{align*}{\frac{ \left ( 1+2\,x \right ) ^{3}}{2} \left ( -{\frac{\sin \left ( 1+2\,x \right ) \cos \left ( 1+2\,x \right ) }{2}}+x+{\frac{1}{2}} \right ) }-{\frac{3\, \left ( \cos \left ( 1+2\,x \right ) \right ) ^{2} \left ( 1+2\,x \right ) ^{2}}{8}}+{\frac{3+6\,x}{4} \left ({\frac{\sin \left ( 1+2\,x \right ) \cos \left ( 1+2\,x \right ) }{2}}+x+{\frac{1}{2}} \right ) }-{\frac{3\, \left ( 1+2\,x \right ) ^{2}}{16}}-{\frac{3\, \left ( \sin \left ( 1+2\,x \right ) \right ) ^{2}}{16}}-{\frac{3\, \left ( 1+2\,x \right ) ^{4}}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983696, size = 69, normalized size = 0.7 \begin{align*} \frac{1}{16} \,{\left (2 \, x + 1\right )}^{4} - \frac{3}{32} \,{\left (2 \,{\left (2 \, x + 1\right )}^{2} - 1\right )} \cos \left (4 \, x + 2\right ) - \frac{1}{16} \,{\left (2 \,{\left (2 \, x + 1\right )}^{3} - 6 \, x - 3\right )} \sin \left (4 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15695, size = 177, normalized size = 1.79 \begin{align*} x^{4} + 2 \, x^{3} - \frac{3}{16} \,{\left (8 \, x^{2} + 8 \, x + 1\right )} \cos \left (2 \, x + 1\right )^{2} - \frac{1}{8} \,{\left (16 \, x^{3} + 24 \, x^{2} + 6 \, x - 1\right )} \cos \left (2 \, x + 1\right ) \sin \left (2 \, x + 1\right ) + \frac{9}{4} \, x^{2} + \frac{5}{4} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.31698, size = 189, normalized size = 1.91 \begin{align*} x^{4} \sin ^{2}{\left (2 x + 1 \right )} + x^{4} \cos ^{2}{\left (2 x + 1 \right )} + 2 x^{3} \sin ^{2}{\left (2 x + 1 \right )} - 2 x^{3} \sin{\left (2 x + 1 \right )} \cos{\left (2 x + 1 \right )} + 2 x^{3} \cos ^{2}{\left (2 x + 1 \right )} + \frac{9 x^{2} \sin ^{2}{\left (2 x + 1 \right )}}{4} - 3 x^{2} \sin{\left (2 x + 1 \right )} \cos{\left (2 x + 1 \right )} + \frac{3 x^{2} \cos ^{2}{\left (2 x + 1 \right )}}{4} + \frac{5 x \sin ^{2}{\left (2 x + 1 \right )}}{4} - \frac{3 x \sin{\left (2 x + 1 \right )} \cos{\left (2 x + 1 \right )}}{4} - \frac{x \cos ^{2}{\left (2 x + 1 \right )}}{4} + \frac{\sin{\left (2 x + 1 \right )} \cos{\left (2 x + 1 \right )}}{8} - \frac{3 \cos ^{2}{\left (2 x + 1 \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0762, size = 78, normalized size = 0.79 \begin{align*} x^{4} + 2 \, x^{3} + \frac{3}{2} \, x^{2} - \frac{3}{32} \,{\left (8 \, x^{2} + 8 \, x + 1\right )} \cos \left (4 \, x + 2\right ) - \frac{1}{16} \,{\left (16 \, x^{3} + 24 \, x^{2} + 6 \, x - 1\right )} \sin \left (4 \, x + 2\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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