Optimal. Leaf size=91 \[ -\frac{\sin ^9(c+d x)}{9 a d}+\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\cos ^8(c+d x)}{8 a d}-\frac{\cos ^6(c+d x)}{6 a d} \]
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Rubi [A] time = 0.160386, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2835, 2565, 14, 2564, 270} \[ -\frac{\sin ^9(c+d x)}{9 a d}+\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\cos ^8(c+d x)}{8 a d}-\frac{\cos ^6(c+d x)}{6 a d} \]
Antiderivative was successfully verified.
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Rule 2835
Rule 2565
Rule 14
Rule 2564
Rule 270
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^5(c+d x) \sin ^3(c+d x) \, dx}{a}-\frac{\int \cos ^5(c+d x) \sin ^4(c+d x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{a d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{a d}\\ &=-\frac{\cos ^6(c+d x)}{6 a d}+\frac{\cos ^8(c+d x)}{8 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^9(c+d x)}{9 a d}\\ \end{align*}
Mathematica [A] time = 0.565633, size = 68, normalized size = 0.75 \[ \frac{\sin ^4(c+d x) \left (-280 \sin ^5(c+d x)+315 \sin ^4(c+d x)+720 \sin ^3(c+d x)-840 \sin ^2(c+d x)-504 \sin (c+d x)+630\right )}{2520 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 69, normalized size = 0.8 \begin{align*}{\frac{1}{da} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{8}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11612, size = 93, normalized size = 1.02 \begin{align*} -\frac{280 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} - 720 \, \sin \left (d x + c\right )^{7} + 840 \, \sin \left (d x + c\right )^{6} + 504 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4}}{2520 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15522, size = 209, normalized size = 2.3 \begin{align*} \frac{315 \, \cos \left (d x + c\right )^{8} - 420 \, \cos \left (d x + c\right )^{6} - 8 \,{\left (35 \, \cos \left (d x + c\right )^{8} - 50 \, \cos \left (d x + c\right )^{6} + 3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{2520 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28428, size = 93, normalized size = 1.02 \begin{align*} -\frac{280 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} - 720 \, \sin \left (d x + c\right )^{7} + 840 \, \sin \left (d x + c\right )^{6} + 504 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4}}{2520 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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