Optimal. Leaf size=22 \[ \frac{(a+b \sin (c+d x))^3}{3 b d} \]
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Rubi [A] time = 0.0269699, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2668, 32} \[ \frac{(a+b \sin (c+d x))^3}{3 b d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^2 \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{(a+b \sin (c+d x))^3}{3 b d}\\ \end{align*}
Mathematica [B] time = 0.0135242, size = 46, normalized size = 2.09 \[ \frac{a^2 \sin (c+d x)}{d}+\frac{a b \sin ^2(c+d x)}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940937, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1854, size = 109, normalized size = 4.95 \begin{align*} -\frac{3 \, a b \cos \left (d x + c\right )^{2} +{\left (b^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.5765, size = 53, normalized size = 2.41 \begin{align*} \begin{cases} \frac{a^{2} \sin{\left (c + d x \right )}}{d} + \frac{a b \sin ^{2}{\left (c + d x \right )}}{d} + \frac{b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{2} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08004, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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