Optimal. Leaf size=139 \[ -\frac{\cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}+\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{5 x}{128 a} \]
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Rubi [A] time = 0.196626, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2568, 2635, 8, 2565, 14} \[ -\frac{\cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}+\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{5 x}{128 a} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2568
Rule 2635
Rule 8
Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{a}-\frac{\int \cos ^6(c+d x) \sin ^3(c+d x) \, dx}{a}\\ &=-\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}+\frac{\int \cos ^6(c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}-\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}+\frac{5 \int \cos ^4(c+d x) \, dx}{48 a}+\frac{\operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}-\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}+\frac{5 \int \cos ^2(c+d x) \, dx}{64 a}\\ &=\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac{5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}-\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}+\frac{5 \int 1 \, dx}{128 a}\\ &=\frac{5 x}{128 a}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac{5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}+\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}-\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}\\ \end{align*}
Mathematica [B] time = 9.54328, size = 479, normalized size = 3.45 \[ -\frac{-5040 d x \sin \left (\frac{c}{2}\right )+1512 \sin \left (\frac{c}{2}+d x\right )-1512 \sin \left (\frac{3 c}{2}+d x\right )-1008 \sin \left (\frac{3 c}{2}+2 d x\right )-1008 \sin \left (\frac{5 c}{2}+2 d x\right )+672 \sin \left (\frac{5 c}{2}+3 d x\right )-672 \sin \left (\frac{7 c}{2}+3 d x\right )+504 \sin \left (\frac{7 c}{2}+4 d x\right )+504 \sin \left (\frac{9 c}{2}+4 d x\right )+336 \sin \left (\frac{11 c}{2}+6 d x\right )+336 \sin \left (\frac{13 c}{2}+6 d x\right )-108 \sin \left (\frac{13 c}{2}+7 d x\right )+108 \sin \left (\frac{15 c}{2}+7 d x\right )+63 \sin \left (\frac{15 c}{2}+8 d x\right )+63 \sin \left (\frac{17 c}{2}+8 d x\right )-28 \sin \left (\frac{17 c}{2}+9 d x\right )+28 \sin \left (\frac{19 c}{2}+9 d x\right )+2520 \cos \left (\frac{c}{2}\right ) (c-2 d x)-1512 \cos \left (\frac{c}{2}+d x\right )-1512 \cos \left (\frac{3 c}{2}+d x\right )-1008 \cos \left (\frac{3 c}{2}+2 d x\right )+1008 \cos \left (\frac{5 c}{2}+2 d x\right )-672 \cos \left (\frac{5 c}{2}+3 d x\right )-672 \cos \left (\frac{7 c}{2}+3 d x\right )+504 \cos \left (\frac{7 c}{2}+4 d x\right )-504 \cos \left (\frac{9 c}{2}+4 d x\right )+336 \cos \left (\frac{11 c}{2}+6 d x\right )-336 \cos \left (\frac{13 c}{2}+6 d x\right )+108 \cos \left (\frac{13 c}{2}+7 d x\right )+108 \cos \left (\frac{15 c}{2}+7 d x\right )+63 \cos \left (\frac{15 c}{2}+8 d x\right )-63 \cos \left (\frac{17 c}{2}+8 d x\right )+28 \cos \left (\frac{17 c}{2}+9 d x\right )+28 \cos \left (\frac{19 c}{2}+9 d x\right )+2520 c \sin \left (\frac{c}{2}\right )-7560 \sin \left (\frac{c}{2}\right )}{129024 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.106, size = 551, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61433, size = 705, normalized size = 5.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19651, size = 220, normalized size = 1.58 \begin{align*} -\frac{896 \, \cos \left (d x + c\right )^{9} - 1152 \, \cos \left (d x + c\right )^{7} - 315 \, d x + 21 \,{\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, 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 = 1.17643, size = 312, normalized size = 2.24 \begin{align*} \frac{\frac{315 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} - 8022 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 16128 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 10458 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 26880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 18270 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 80640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 48384 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 18270 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 48384 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 10458 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6912 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 8022 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2304 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{9} a}}{8064 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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