Optimal. Leaf size=73 \[ \frac{\sin ^6(c+d x)}{6 a d}-\frac{\sin ^5(c+d x)}{5 a d}-\frac{\sin ^4(c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.155688, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2835, 2564, 14} \[ \frac{\sin ^6(c+d x)}{6 a d}-\frac{\sin ^5(c+d x)}{5 a d}-\frac{\sin ^4(c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^3(c+d x) \sin ^2(c+d x) \, dx}{a}-\frac{\int \cos ^3(c+d x) \sin ^3(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^4(c+d x)}{4 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^6(c+d x)}{6 a d}\\ \end{align*}
Mathematica [A] time = 0.207874, size = 48, normalized size = 0.66 \[ \frac{\sin ^3(c+d x) \left (10 \sin ^3(c+d x)-12 \sin ^2(c+d x)-15 \sin (c+d x)+20\right )}{60 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{da} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13324, size = 66, normalized size = 0.9 \begin{align*} \frac{10 \, \sin \left (d x + c\right )^{6} - 12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3}}{60 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66327, size = 149, normalized size = 2.04 \begin{align*} -\frac{10 \, \cos \left (d x + c\right )^{6} - 15 \, \cos \left (d x + c\right )^{4} + 4 \,{\left (3 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{60 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 79.2048, size = 862, normalized size = 11.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14886, size = 66, normalized size = 0.9 \begin{align*} \frac{10 \, \sin \left (d x + c\right )^{6} - 12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3}}{60 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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