Optimal. Leaf size=33 \[ \frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+x)}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int x^2 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 50, normalized size = 1.52 \[ \frac {3 \, a \cos \left (d x + c\right )^{4} - 6 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 28, normalized size = 0.85 \[ \frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 28, normalized size = 0.85 \[ \frac {\frac {\left (\sin ^{4}\left (d x +c \right )\right ) a}{4}+\frac {\left (\sin ^{3}\left (d x +c \right )\right ) a}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 28, normalized size = 0.85 \[ \frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.73 \[ \frac {a\,{\sin \left (c+d\,x\right )}^3\,\left (3\,\sin \left (c+d\,x\right )+4\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.31, size = 42, normalized size = 1.27 \[ \begin {cases} \frac {a \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac {a \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{2}{\relax (c )} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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