Optimal. Leaf size=37 \[ \frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.08, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2835, 2564, 30} \[ \frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2835
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos (c+d x) \sin (c+d x) \, dx}{a}-\frac {\int \cos (c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}(\int x \, dx,x,\sin (c+d x))}{a d}-\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 28, normalized size = 0.76 \[ \frac {(3-2 \sin (c+d x)) \sin ^2(c+d x)}{6 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 37, normalized size = 1.00 \[ -\frac {3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 29, normalized size = 0.78 \[ -\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2}}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 30, normalized size = 0.81 \[ -\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 0.78 \[ -\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2}}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.54, size = 26, normalized size = 0.70 \[ -\frac {{\sin \left (c+d\,x\right )}^2\,\left (2\,\sin \left (c+d\,x\right )-3\right )}{6\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.74, size = 224, normalized size = 6.05 \[ \begin {cases} \frac {6 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} - \frac {8 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} + \frac {6 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\relax (c )} \cos ^{3}{\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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