Optimal. Leaf size=33 \[ \frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2833, 12, 43} \[ \frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+x)}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}(\int x (a+x) \, dx,x,a \sin (c+d x))}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 30, normalized size = 0.91 \[ \frac {4 a \sin ^3(c+d x)-3 a \cos (2 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 39, normalized size = 1.18 \[ -\frac {3 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 28, normalized size = 0.85 \[ \frac {2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.85 \[ \frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right ) a}{3}+\frac {\left (\sin ^{2}\left (d x +c \right )\right ) a}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 28, normalized size = 0.85 \[ \frac {2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.73 \[ \frac {a\,{\sin \left (c+d\,x\right )}^2\,\left (2\,\sin \left (c+d\,x\right )+3\right )}{6\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 41, normalized size = 1.24 \[ \begin {cases} \frac {a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sin ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin {\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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