Optimal. Leaf size=19 \[ a x+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x) \]
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Rubi [A] time = 0.0092947, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ a x+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \sin ^2(x)\right ) \, dx &=a x+b \int \sin ^2(x) \, dx\\ &=a x-\frac{1}{2} b \cos (x) \sin (x)+\frac{1}{2} b \int 1 \, dx\\ &=a x+\frac{b x}{2}-\frac{1}{2} b \cos (x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0033692, size = 19, normalized size = 1. \[ a x+\frac{b x}{2}-\frac{1}{4} b \sin (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 17, normalized size = 0.9 \begin{align*} ax+b \left ( -{\frac{\sin \left ( x \right ) \cos \left ( x \right ) }{2}}+{\frac{x}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942653, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{4} \, b{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58573, size = 54, normalized size = 2.84 \begin{align*} -\frac{1}{2} \, b \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2} \,{\left (2 \, a + b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099959, size = 15, normalized size = 0.79 \begin{align*} a x + b \left (\frac{x}{2} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13251, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{4} \, b{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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