Optimal. Leaf size=159 \[ -\frac {\cos ^9(c+d x)}{9 a^2 d}+\frac {3 \cos ^7(c+d x)}{7 a^2 d}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{4 a^2 d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{32 a^2 d}-\frac {3 \sin (c+d x) \cos (c+d x)}{64 a^2 d}-\frac {3 x}{64 a^2} \]
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Rubi [A] time = 0.36, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac {\cos ^9(c+d x)}{9 a^2 d}+\frac {3 \cos ^7(c+d x)}{7 a^2 d}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{4 a^2 d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{32 a^2 d}-\frac {3 \sin (c+d x) \cos (c+d x)}{64 a^2 d}-\frac {3 x}{64 a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 270
Rule 2565
Rule 2568
Rule 2635
Rule 2873
Rule 2875
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cos ^4(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cos ^4(c+d x) \sin ^3(c+d x)-2 a^2 \cos ^4(c+d x) \sin ^4(c+d x)+a^2 \cos ^4(c+d x) \sin ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cos ^4(c+d x) \sin ^3(c+d x) \, dx}{a^2}+\frac {\int \cos ^4(c+d x) \sin ^5(c+d x) \, dx}{a^2}-\frac {2 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx}{a^2}\\ &=\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{4 a^2 d}-\frac {3 \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx}{4 a^2}-\frac {\operatorname {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {\cos ^5(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{4 a^2 d}-\frac {\int \cos ^4(c+d x) \, dx}{8 a^2}-\frac {\operatorname {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {3 \cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^9(c+d x)}{9 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{32 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{4 a^2 d}-\frac {3 \int \cos ^2(c+d x) \, dx}{32 a^2}\\ &=-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {3 \cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^9(c+d x)}{9 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{32 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{4 a^2 d}-\frac {3 \int 1 \, dx}{64 a^2}\\ &=-\frac {3 x}{64 a^2}-\frac {2 \cos ^5(c+d x)}{5 a^2 d}+\frac {3 \cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^9(c+d x)}{9 a^2 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{64 a^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{32 a^2 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^5(c+d x) \sin ^3(c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [B] time = 6.91, size = 430, normalized size = 2.70 \[ -\frac {15120 d x \sin \left (\frac {c}{2}\right )-11340 \sin \left (\frac {c}{2}+d x\right )+11340 \sin \left (\frac {3 c}{2}+d x\right )-3360 \sin \left (\frac {5 c}{2}+3 d x\right )+3360 \sin \left (\frac {7 c}{2}+3 d x\right )-2520 \sin \left (\frac {7 c}{2}+4 d x\right )-2520 \sin \left (\frac {9 c}{2}+4 d x\right )+1008 \sin \left (\frac {9 c}{2}+5 d x\right )-1008 \sin \left (\frac {11 c}{2}+5 d x\right )+450 \sin \left (\frac {13 c}{2}+7 d x\right )-450 \sin \left (\frac {15 c}{2}+7 d x\right )+315 \sin \left (\frac {15 c}{2}+8 d x\right )+315 \sin \left (\frac {17 c}{2}+8 d x\right )-70 \sin \left (\frac {17 c}{2}+9 d x\right )+70 \sin \left (\frac {19 c}{2}+9 d x\right )+420 \cos \left (\frac {c}{2}\right ) (330 c+36 d x+7)+11340 \cos \left (\frac {c}{2}+d x\right )+11340 \cos \left (\frac {3 c}{2}+d x\right )+3360 \cos \left (\frac {5 c}{2}+3 d x\right )+3360 \cos \left (\frac {7 c}{2}+3 d x\right )-2520 \cos \left (\frac {7 c}{2}+4 d x\right )+2520 \cos \left (\frac {9 c}{2}+4 d x\right )-1008 \cos \left (\frac {9 c}{2}+5 d x\right )-1008 \cos \left (\frac {11 c}{2}+5 d x\right )-450 \cos \left (\frac {13 c}{2}+7 d x\right )-450 \cos \left (\frac {15 c}{2}+7 d x\right )+315 \cos \left (\frac {15 c}{2}+8 d x\right )-315 \cos \left (\frac {17 c}{2}+8 d x\right )+70 \cos \left (\frac {17 c}{2}+9 d x\right )+70 \cos \left (\frac {19 c}{2}+9 d x\right )+138600 c \sin \left (\frac {c}{2}\right )-78960 \sin \left (\frac {c}{2}\right )}{322560 a^2 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 90, normalized size = 0.57 \[ -\frac {2240 \, \cos \left (d x + c\right )^{9} - 8640 \, \cos \left (d x + c\right )^{7} + 8064 \, \cos \left (d x + c\right )^{5} + 945 \, d x + 315 \, {\left (16 \, \cos \left (d x + c\right )^{7} - 24 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 231, normalized size = 1.45 \[ -\frac {\frac {945 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 40320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 147840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 120960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 330624 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 106470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 97650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 19584 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14976 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1664\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{9} a^{2}}}{20160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 551, normalized size = 3.47 \[ -\frac {52}{315 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {52 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {68 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {155 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {169 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {164 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {12 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {169 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {44 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {155 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {4 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {13 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {3 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 542, normalized size = 3.41 \[ \frac {\frac {\frac {945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14976 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8190 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {19584 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {97650 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8064 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {106470 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {330624 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {120960 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {106470 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {147840 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {97650 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {40320 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {8190 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {945 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - 1664}{a^{2} + \frac {9 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {36 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {126 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {126 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {84 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {36 \, a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {9 \, a^{2} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {a^{2} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}}} - \frac {945 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{10080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.68, size = 225, normalized size = 1.42 \[ -\frac {3\,x}{64\,a^2}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {164\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}-\frac {169\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {68\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}+\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}+\frac {52}{315}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
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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