Optimal. Leaf size=77 \[ \frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {5 x}{2 a^3}+\frac {2 \cos ^5(c+d x)}{a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2680, 2682, 2635, 8} \[ \frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {5 x}{2 a^3}+\frac {2 \cos ^5(c+d x)}{a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2680
Rule 2682
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {5 \int \frac {\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {5 \int \cos ^2(c+d x) \, dx}{a^3}\\ &=\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}+\frac {5 \int 1 \, dx}{2 a^3}\\ &=\frac {5 x}{2 a^3}+\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos ^5(c+d x)}{a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 121, normalized size = 1.57 \[ -\frac {\left (30 \sqrt {1-\sin (c+d x)} \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )+\sqrt {\sin (c+d x)+1} \left (2 \sin ^3(c+d x)-11 \sin ^2(c+d x)+31 \sin (c+d x)-22\right )\right ) \cos ^7(c+d x)}{6 a^3 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 45, normalized size = 0.58 \[ -\frac {2 \, \cos \left (d x + c\right )^{3} - 15 \, d x + 9 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 24 \, \cos \left (d x + c\right )}{6 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 88, normalized size = 1.14 \[ \frac {\frac {15 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 22\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 177, normalized size = 2.30 \[ \frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {22}{3 a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 184, normalized size = 2.39 \[ -\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {18 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {9 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 22}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.64, size = 57, normalized size = 0.74 \[ \frac {5\,x}{2\,a^3}+\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}-\frac {3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 108.27, size = 690, normalized size = 8.96 \[ \begin {cases} \frac {15 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {45 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {45 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {15 d x}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {18 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {36 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {96 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} - \frac {18 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} + \frac {44}{6 a^{3} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 18 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{6}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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