Optimal. Leaf size=38 \[ \frac {a \sin (c+d x)}{d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2734} \[ \frac {a \sin (c+d x)}{d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2} \]
Antiderivative was successfully verified.
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Rule 2734
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x)) \, dx &=\frac {a x}{2}+\frac {a \sin (c+d x)}{d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 32, normalized size = 0.84 \[ \frac {a (2 (c+d x)+4 \sin (c+d x)+\sin (2 (c+d x)))}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.33, size = 29, normalized size = 0.76 \[ \frac {a d x + {\left (a \cos \left (d x + c\right ) + 2 \, a\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 31, normalized size = 0.82 \[ \frac {1}{2} \, a x + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 1.00 \[ \frac {a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\sin \left (d x +c \right ) a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 34, normalized size = 0.89 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a + 4 \, a \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 50, normalized size = 1.32 \[ \frac {a\,x}{2}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 66, normalized size = 1.74 \[ \begin {cases} \frac {a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {a \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + a\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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