Optimal. Leaf size=43 \[ -\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2669, 2635, 8} \[ -\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx &=-\frac {a \cos ^3(c+d x)}{3 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a \int 1 \, dx\\ &=\frac {a x}{2}-\frac {a \cos ^3(c+d x)}{3 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 = 46, normalized size = 1.07 \[ \frac {a (c+d x)}{2 d}+\frac {a \sin (2 (c+d x))}{4 d}-\frac {a \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 37, normalized size = 0.86 \[ -\frac {2 \, a \cos \left (d x + c\right )^{3} - 3 \, a d x - 3 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 47, normalized size = 1.09 \[ \frac {1}{2} \, a x - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {a \cos \left (d x + c\right )}{4 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 41, normalized size = 0.95 \[ \frac {-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) a}{3}+a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 37, normalized size = 0.86 \[ -\frac {4 \, a \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.76, size = 103, normalized size = 2.40 \[ \frac {a\,x}{2}+\frac {-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {a\,\left (9\,c+9\,d\,x-12\right )}{6}-\frac {3\,a\,\left (c+d\,x\right )}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a\,\left (3\,c+3\,d\,x-4\right )}{6}-\frac {a\,\left (c+d\,x\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 71, normalized size = 1.65 \[ \begin {cases} \frac {a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {a \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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