Optimal. Leaf size=37 \[ \frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^4(c+d x)}{4 a d} \]
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Rubi [A] time = 0.10, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ \frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x) x^2}{a^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x) x^2 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^4(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 28, normalized size = 0.76 \[ \frac {(4-3 \sin (c+d x)) \sin ^3(c+d x)}{12 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 47, normalized size = 1.27 \[ -\frac {3 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.78 \[ -\frac {3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3}}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 30, normalized size = 0.81 \[ -\frac {\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 29, normalized size = 0.78 \[ -\frac {3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3}}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.70 \[ -\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,\sin \left (c+d\,x\right )-4\right )}{12\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.00, size = 277, normalized size = 7.49 \[ \begin {cases} \frac {8 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} - \frac {12 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} + \frac {8 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 12 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\relax (c )} \cos ^{3}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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