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 = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 8
Rule 2635
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*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ a x+\frac {b x}{2}-\frac {1}{4} b \sin (2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 16, normalized size = 0.84 \[ -\frac {1}{2} \, b \cos \relax (x) \sin \relax (x) + \frac {1}{2} \, {\left (2 \, a + b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 17, normalized size = 0.89 \[ \frac {1}{4} \, b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 17, normalized size = 0.89 \[ a x +b \left (-\frac {\sin \relax (x ) \cos \relax (x )}{2}+\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 17, normalized size = 0.89 \[ \frac {1}{4} \, b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.31, size = 15, normalized size = 0.79 \[ x\,\left (a+\frac {b}{2}\right )-\frac {b\,\sin \left (2\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 15, normalized size = 0.79 \[ a x + b \left (\frac {x}{2} - \frac {\sin {\relax (x )} \cos {\relax (x )}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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