Optimal. Leaf size=165 \[ \frac{\cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac{3 x}{256 a} \]
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Rubi [A] time = 0.237026, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2565, 14, 2568, 2635, 8} \[ \frac{\cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac{3 x}{256 a} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2565
Rule 14
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^6(c+d x) \sin ^3(c+d x) \, dx}{a}-\frac{\int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{a}\\ &=\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac{3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{10 a}-\frac{\operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac{3 \int \cos ^6(c+d x) \, dx}{80 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{\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac{\int \cos ^4(c+d x) \, dx}{32 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\cos ^9(c+d x)}{9 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac{3 \int \cos ^2(c+d x) \, dx}{128 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\cos ^9(c+d x)}{9 a d}-\frac{3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac{3 \int 1 \, dx}{256 a}\\ &=-\frac{3 x}{256 a}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\cos ^9(c+d x)}{9 a d}-\frac{3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac{3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac{\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}\\ \end{align*}
Mathematica [B] time = 15.1624, size = 533, normalized size = 3.23 \[ -\frac{15120 d x \sin \left (\frac{c}{2}\right )-15120 \sin \left (\frac{c}{2}+d x\right )+15120 \sin \left (\frac{3 c}{2}+d x\right )+1260 \sin \left (\frac{3 c}{2}+2 d x\right )+1260 \sin \left (\frac{5 c}{2}+2 d x\right )-6720 \sin \left (\frac{5 c}{2}+3 d x\right )+6720 \sin \left (\frac{7 c}{2}+3 d x\right )-2520 \sin \left (\frac{7 c}{2}+4 d x\right )-2520 \sin \left (\frac{9 c}{2}+4 d x\right )-630 \sin \left (\frac{11 c}{2}+6 d x\right )-630 \sin \left (\frac{13 c}{2}+6 d x\right )+1080 \sin \left (\frac{13 c}{2}+7 d x\right )-1080 \sin \left (\frac{15 c}{2}+7 d x\right )+315 \sin \left (\frac{15 c}{2}+8 d x\right )+315 \sin \left (\frac{17 c}{2}+8 d x\right )+280 \sin \left (\frac{17 c}{2}+9 d x\right )-280 \sin \left (\frac{19 c}{2}+9 d x\right )+126 \sin \left (\frac{19 c}{2}+10 d x\right )+126 \sin \left (\frac{21 c}{2}+10 d x\right )-1260 \cos \left (\frac{c}{2}\right ) (25 c-12 d x)+15120 \cos \left (\frac{c}{2}+d x\right )+15120 \cos \left (\frac{3 c}{2}+d x\right )+1260 \cos \left (\frac{3 c}{2}+2 d x\right )-1260 \cos \left (\frac{5 c}{2}+2 d x\right )+6720 \cos \left (\frac{5 c}{2}+3 d x\right )+6720 \cos \left (\frac{7 c}{2}+3 d x\right )-2520 \cos \left (\frac{7 c}{2}+4 d x\right )+2520 \cos \left (\frac{9 c}{2}+4 d x\right )-630 \cos \left (\frac{11 c}{2}+6 d x\right )+630 \cos \left (\frac{13 c}{2}+6 d x\right )-1080 \cos \left (\frac{13 c}{2}+7 d x\right )-1080 \cos \left (\frac{15 c}{2}+7 d x\right )+315 \cos \left (\frac{15 c}{2}+8 d x\right )-315 \cos \left (\frac{17 c}{2}+8 d x\right )-280 \cos \left (\frac{17 c}{2}+9 d x\right )-280 \cos \left (\frac{19 c}{2}+9 d x\right )+126 \cos \left (\frac{19 c}{2}+10 d x\right )-126 \cos \left (\frac{21 c}{2}+10 d x\right )-31500 c \sin \left (\frac{c}{2}\right )+37800 \sin \left (\frac{c}{2}\right )}{1290240 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.11, size = 619, normalized size = 3.8 \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 = 1.75912, size = 787, normalized size = 4.77 \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 = 1.17474, size = 252, normalized size = 1.53 \begin{align*} \frac{8960 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} - 945 \, d x - 63 \,{\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \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 )}{80640 \, 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 = 1.29525, size = 347, normalized size = 2.1 \begin{align*} -\frac{\frac{945 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (945 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{19} + 9135 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} + 161280 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} - 218484 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 107520 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 653940 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 537600 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 1183770 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 322560 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 1183770 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 653940 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 414720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 218484 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 46080 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 9135 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 25600 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2560\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{10} a}}{80640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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