Optimal. Leaf size=105 \[ \frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac {13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {13 x}{8 a^3} \]
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Rubi [A] time = 0.22, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2869, 2757, 2635, 8, 2633} \[ \frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac {13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {13 x}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2757
Rule 2869
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \sin ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (a^3 \sin ^2(c+d x)-3 a^3 \sin ^3(c+d x)+3 a^3 \sin ^4(c+d x)-a^3 \sin ^5(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \sin ^2(c+d x) \, dx}{a^3}-\frac {\int \sin ^5(c+d x) \, dx}{a^3}-\frac {3 \int \sin ^3(c+d x) \, dx}{a^3}+\frac {3 \int \sin ^4(c+d x) \, dx}{a^3}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {\int 1 \, dx}{2 a^3}+\frac {9 \int \sin ^2(c+d x) \, dx}{4 a^3}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac {x}{2 a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}+\frac {9 \int 1 \, dx}{8 a^3}\\ &=\frac {13 x}{8 a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d}\\ \end {align*}
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Mathematica [B] time = 1.77, size = 310, normalized size = 2.95 \[ \frac {1560 d x \sin \left (\frac {c}{2}\right )-1380 \sin \left (\frac {c}{2}+d x\right )+1380 \sin \left (\frac {3 c}{2}+d x\right )-480 \sin \left (\frac {3 c}{2}+2 d x\right )-480 \sin \left (\frac {5 c}{2}+2 d x\right )+170 \sin \left (\frac {5 c}{2}+3 d x\right )-170 \sin \left (\frac {7 c}{2}+3 d x\right )+45 \sin \left (\frac {7 c}{2}+4 d x\right )+45 \sin \left (\frac {9 c}{2}+4 d x\right )-6 \sin \left (\frac {9 c}{2}+5 d x\right )+6 \sin \left (\frac {11 c}{2}+5 d x\right )+1560 d x \cos \left (\frac {c}{2}\right )+1380 \cos \left (\frac {c}{2}+d x\right )+1380 \cos \left (\frac {3 c}{2}+d x\right )-480 \cos \left (\frac {3 c}{2}+2 d x\right )+480 \cos \left (\frac {5 c}{2}+2 d x\right )-170 \cos \left (\frac {5 c}{2}+3 d x\right )-170 \cos \left (\frac {7 c}{2}+3 d x\right )+45 \cos \left (\frac {7 c}{2}+4 d x\right )-45 \cos \left (\frac {9 c}{2}+4 d x\right )+6 \cos \left (\frac {9 c}{2}+5 d x\right )+6 \cos \left (\frac {11 c}{2}+5 d x\right )+10 \sin \left (\frac {c}{2}\right )}{960 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 68, normalized size = 0.65 \[ \frac {24 \, \cos \left (d x + c\right )^{5} - 200 \, \cos \left (d x + c\right )^{3} + 195 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{120 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 127, normalized size = 1.21 \[ \frac {\frac {195 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 304\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 279, normalized size = 2.66 \[ \frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {12 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {116 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {76 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {76}{15 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.47, size = 290, normalized size = 2.76 \[ -\frac {\frac {\frac {195 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1520 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {195 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 304}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {195 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.03, size = 95, normalized size = 0.90 \[ \frac {13\,x}{8\,a^3}+\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {5\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}+\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}-\frac {19\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \]
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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