Optimal. Leaf size=73 \[ \frac {\sin (a+b x) \cos (a+b x)}{4 b^3}+\frac {x \sin ^2(a+b x)}{2 b^2}-\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}-\frac {x}{4 b^2}+\frac {x^3}{6} \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3311, 30, 2635, 8} \[ \frac {x \sin ^2(a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{4 b^3}-\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}-\frac {x}{4 b^2}+\frac {x^3}{6} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2635
Rule 3311
Rubi steps
\begin {align*} \int x^2 \sin ^2(a+b x) \, dx &=-\frac {x^2 \cos (a+b x) \sin (a+b x)}{2 b}+\frac {x \sin ^2(a+b x)}{2 b^2}+\frac {\int x^2 \, dx}{2}-\frac {\int \sin ^2(a+b x) \, dx}{2 b^2}\\ &=\frac {x^3}{6}+\frac {\cos (a+b x) \sin (a+b x)}{4 b^3}-\frac {x^2 \cos (a+b x) \sin (a+b x)}{2 b}+\frac {x \sin ^2(a+b x)}{2 b^2}-\frac {\int 1 \, dx}{4 b^2}\\ &=-\frac {x}{4 b^2}+\frac {x^3}{6}+\frac {\cos (a+b x) \sin (a+b x)}{4 b^3}-\frac {x^2 \cos (a+b x) \sin (a+b x)}{2 b}+\frac {x \sin ^2(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 47, normalized size = 0.64 \[ \frac {\left (3-6 b^2 x^2\right ) \sin (2 (a+b x))-6 b x \cos (2 (a+b x))+4 b^3 x^3}{24 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 54, normalized size = 0.74 \[ \frac {2 \, b^{3} x^{3} - 6 \, b x \cos \left (b x + a\right )^{2} - 3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, b x}{12 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 45, normalized size = 0.62 \[ \frac {1}{6} \, x^{3} - \frac {x \cos \left (2 \, b x + 2 \, a\right )}{4 \, b^{2}} - \frac {{\left (2 \, b^{2} x^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 158, normalized size = 2.16 \[ \frac {\left (\frac {b x}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {a}{2}\right ) a^{2}+\frac {b x}{4}-\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}+\frac {a}{4}-2 \left (\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}+\left (b x +a \right ) \left (\frac {b x}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}\right ) a +\left (b x +a \right )^{2} \left (\frac {b x}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{3}}{3}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 117, normalized size = 1.60 \[ \frac {4 \, {\left (b x + a\right )}^{3} + 6 \, {\left (2 \, b x + 2 \, a - \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} - 6 \, {\left (2 \, {\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right )\right )} a - 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 52, normalized size = 0.71 \[ \frac {x^3}{6}+\frac {\sin \left (2\,a+2\,b\,x\right )}{8\,b^3}-\frac {x\,\cos \left (2\,a+2\,b\,x\right )}{4\,b^2}-\frac {x^2\,\sin \left (2\,a+2\,b\,x\right )}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.13, size = 105, normalized size = 1.44 \[ \begin {cases} \frac {x^{3} \sin ^{2}{\left (a + b x \right )}}{6} + \frac {x^{3} \cos ^{2}{\left (a + b x \right )}}{6} - \frac {x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {x \sin ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {x \cos ^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sin ^{2}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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