Optimal. Leaf size=22 \[ \frac {(a \sin (c+d x)+a)^2}{2 a d} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2667} \[ \frac {a \sin ^2(c+d x)}{2 d}+\frac {a \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2667
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}(\int (a+x) \, dx,x,a \sin (c+d x))}{a d}\\ &=\frac {a \sin (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.77 \[ -\frac {a \cos ^2(c+d x)}{2 d}+\frac {a \sin (c) \cos (d x)}{d}+\frac {a \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 25, normalized size = 1.14 \[ -\frac {a \cos \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 25, normalized size = 1.14 \[ \frac {a \sin \left (d x + c\right )^{2} + 2 \, a \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.14 \[ \frac {\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2}+a \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 20, normalized size = 0.91 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 0.91 \[ \frac {a\,\sin \left (c+d\,x\right )\,\left (\sin \left (c+d\,x\right )+2\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 34, normalized size = 1.55 \[ \begin {cases} \frac {a \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac {a \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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