Optimal. Leaf size=45 \[ \frac {2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac {(a \sin (c+d x)+a)^4}{4 a^3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 43} \[ \frac {2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac {(a \sin (c+d x)+a)^4}{4 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x) (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (a+x)^2-(a+x)^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {2 (a+a \sin (c+d x))^3}{3 a^2 d}-\frac {(a+a \sin (c+d x))^4}{4 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.98 \[ -\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}-\frac {a \cos ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 39, normalized size = 0.87 \[ -\frac {3 \, a \cos \left (d x + c\right )^{4} - 4 \, {\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 48, normalized size = 1.07 \[ -\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 6 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 36, normalized size = 0.80 \[ \frac {-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) a}{4}+\frac {a \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 48, normalized size = 1.07 \[ -\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 6 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 46, normalized size = 1.02 \[ \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+a\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 60, normalized size = 1.33 \[ \begin {cases} \frac {2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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