Optimal. Leaf size=30 \[ a x-\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2} \]
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Rubi [A] time = 0.0148417, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2635, 8} \[ a x-\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \sin ^2(c+d x)\right ) \, dx &=a x+b \int \sin ^2(c+d x) \, dx\\ &=a x-\frac{b \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} b \int 1 \, dx\\ &=a x+\frac{b x}{2}-\frac{b \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0282181, size = 33, normalized size = 1.1 \[ a x+\frac{b (c+d x)}{2 d}-\frac{b \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 32, normalized size = 1.1 \begin{align*} ax+{\frac{b}{d} \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944351, size = 39, normalized size = 1.3 \begin{align*} a x + \frac{{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} b}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57504, size = 72, normalized size = 2.4 \begin{align*} \frac{{\left (2 \, a + b\right )} d x - b \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.429219, size = 51, normalized size = 1.7 \begin{align*} a x + b \left (\begin{cases} \frac{x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{x \cos ^{2}{\left (c + d x \right )}}{2} - \frac{\sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \sin ^{2}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12648, size = 34, normalized size = 1.13 \begin{align*} \frac{1}{4} \, b{\left (2 \, x - \frac{\sin \left (2 \, d x + 2 \, c\right )}{d}\right )} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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