Optimal. Leaf size=103 \[ \frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {7 x}{8 a^3}+\frac {2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2680, 2682, 2635, 8} \[ \frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac {7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {7 x}{8 a^3}+\frac {2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2680
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int \frac {\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{3 a^2}\\ &=\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int \cos ^4(c+d x) \, dx}{3 a^3}\\ &=\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int \cos ^2(c+d x) \, dx}{4 a^3}\\ &=\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}+\frac {7 \int 1 \, dx}{8 a^3}\\ &=\frac {7 x}{8 a^3}+\frac {7 \cos ^5(c+d x)}{15 a^3 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{12 a^3 d}+\frac {2 \cos ^7(c+d x)}{3 a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 141, normalized size = 1.37 \[ -\frac {\left (\sqrt {\sin (c+d x)+1} \left (24 \sin ^5(c+d x)-114 \sin ^4(c+d x)+202 \sin ^3(c+d x)-127 \sin ^2(c+d x)-121 \sin (c+d x)+136\right )-210 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}\right ) \cos ^9(c+d x)}{120 a^3 d (\sin (c+d x)-1)^5 (\sin (c+d x)+1)^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 60, normalized size = 0.58 \[ -\frac {24 \, \cos \left (d x + c\right )^{5} - 160 \, \cos \left (d x + c\right )^{3} - 105 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 140, normalized size = 1.36 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 136\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 313, normalized size = 3.04 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {6 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {13 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {16 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {20 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {34}{15 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.77, size = 310, normalized size = 3.01 \[ \frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {390 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 136}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 81, normalized size = 0.79 \[ \frac {7\,x}{8\,a^3}+\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}-\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}+\frac {7\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \]
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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