Optimal. Leaf size=209 \[ -\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac {5 x}{1024 a} \]
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Rubi [A] time = 0.28, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2839, 2565, 270, 2568, 2635, 8} \[ -\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac {\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac {5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac {5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac {5 x}{1024 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \cos ^6(c+d x) \sin ^5(c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^6(c+d x) \, dx}{a}\\ &=\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{12 a}-\frac {\operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{8 a}-\frac {\operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {\int \cos ^6(c+d x) \, dx}{64 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int \cos ^4(c+d x) \, dx}{384 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int \cos ^2(c+d x) \, dx}{512 a}\\ &=-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}-\frac {5 \int 1 \, dx}{1024 a}\\ &=-\frac {5 x}{1024 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {2 \cos ^9(c+d x)}{9 a d}-\frac {\cos ^{11}(c+d x)}{11 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{1024 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{1536 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{384 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{64 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{24 a d}+\frac {\cos ^7(c+d x) \sin ^5(c+d x)}{12 a d}\\ \end {align*}
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Mathematica [B] time = 14.28, size = 518, normalized size = 2.48 \[ -\frac {55440 d x \sin \left (\frac {c}{2}\right )-55440 \sin \left (\frac {c}{2}+d x\right )+55440 \sin \left (\frac {3 c}{2}+d x\right )-18480 \sin \left (\frac {5 c}{2}+3 d x\right )+18480 \sin \left (\frac {7 c}{2}+3 d x\right )-10395 \sin \left (\frac {7 c}{2}+4 d x\right )-10395 \sin \left (\frac {9 c}{2}+4 d x\right )+5544 \sin \left (\frac {9 c}{2}+5 d x\right )-5544 \sin \left (\frac {11 c}{2}+5 d x\right )+3960 \sin \left (\frac {13 c}{2}+7 d x\right )-3960 \sin \left (\frac {15 c}{2}+7 d x\right )+2079 \sin \left (\frac {15 c}{2}+8 d x\right )+2079 \sin \left (\frac {17 c}{2}+8 d x\right )-616 \sin \left (\frac {17 c}{2}+9 d x\right )+616 \sin \left (\frac {19 c}{2}+9 d x\right )-504 \sin \left (\frac {21 c}{2}+11 d x\right )+504 \sin \left (\frac {23 c}{2}+11 d x\right )-231 \sin \left (\frac {23 c}{2}+12 d x\right )-231 \sin \left (\frac {25 c}{2}+12 d x\right )+55440 d x \cos \left (\frac {c}{2}\right )+55440 \cos \left (\frac {c}{2}+d x\right )+55440 \cos \left (\frac {3 c}{2}+d x\right )+18480 \cos \left (\frac {5 c}{2}+3 d x\right )+18480 \cos \left (\frac {7 c}{2}+3 d x\right )-10395 \cos \left (\frac {7 c}{2}+4 d x\right )+10395 \cos \left (\frac {9 c}{2}+4 d x\right )-5544 \cos \left (\frac {9 c}{2}+5 d x\right )-5544 \cos \left (\frac {11 c}{2}+5 d x\right )-3960 \cos \left (\frac {13 c}{2}+7 d x\right )-3960 \cos \left (\frac {15 c}{2}+7 d x\right )+2079 \cos \left (\frac {15 c}{2}+8 d x\right )-2079 \cos \left (\frac {17 c}{2}+8 d x\right )+616 \cos \left (\frac {17 c}{2}+9 d x\right )+616 \cos \left (\frac {19 c}{2}+9 d x\right )+504 \cos \left (\frac {21 c}{2}+11 d x\right )+504 \cos \left (\frac {23 c}{2}+11 d x\right )-231 \cos \left (\frac {23 c}{2}+12 d x\right )+231 \cos \left (\frac {25 c}{2}+12 d x\right )+99792 \sin \left (\frac {c}{2}\right )}{11354112 a d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 110, normalized size = 0.53 \[ -\frac {64512 \, \cos \left (d x + c\right )^{11} - 157696 \, \cos \left (d x + c\right )^{9} + 101376 \, \cos \left (d x + c\right )^{7} + 3465 \, d x - 231 \, {\left (256 \, \cos \left (d x + c\right )^{11} - 640 \, \cos \left (d x + c\right )^{9} + 432 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{709632 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 309, normalized size = 1.48 \[ -\frac {\frac {3465 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{23} + 40425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{21} + 215523 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 3784704 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} - 5794173 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 5677056 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 19523658 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11354112 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 35058870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 3784704 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 35058870 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4866048 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 19523658 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9732096 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 5794173 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1982464 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 215523 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 540672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 98304 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8192\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{12} a}}{709632 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 755, normalized size = 3.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 705, normalized size = 3.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.85, size = 303, normalized size = 1.45 \[ -\frac {5\,x}{1024\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{23}}{512}+\frac {175\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{1536}+\frac {311\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{512}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{3}-\frac {8361\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{512}-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {42259\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{768}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {25295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{256}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {25295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{256}-\frac {96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{7}-\frac {42259\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{768}+\frac {192\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}+\frac {8361\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512}-\frac {352\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{63}-\frac {311\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{512}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{21}-\frac {175\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{231}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}+\frac {16}{693}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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