Optimal. Leaf size=25 \[ \frac{x}{2}-\frac{\sin (a+b x) \cos (a+b x)}{2 b} \]
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Rubi [A] time = 0.008774, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{x}{2}-\frac{\sin (a+b x) \cos (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^2(a+b x) \, dx &=-\frac{\cos (a+b x) \sin (a+b x)}{2 b}+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}-\frac{\cos (a+b x) \sin (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0283056, size = 23, normalized size = 0.92 \[ -\frac{\sin (2 (a+b x))-2 (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 27, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995329, size = 32, normalized size = 1.28 \begin{align*} \frac{2 \, b x + 2 \, a - \sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52304, size = 55, normalized size = 2.2 \begin{align*} \frac{b x - \cos \left (b x + a\right ) \sin \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.209137, size = 46, normalized size = 1.84 \begin{align*} \begin{cases} \frac{x \sin ^{2}{\left (a + b x \right )}}{2} + \frac{x \cos ^{2}{\left (a + b x \right )}}{2} - \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13473, size = 24, normalized size = 0.96 \begin{align*} \frac{1}{2} \, x - \frac{\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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