Optimal. Leaf size=27 \[ -\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2671} \[ -\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.04 \[ -\frac {\cos ^3(c+d x)}{3 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 95, normalized size = 3.52 \[ -\frac {\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 36, normalized size = 1.33 \[ -\frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{3 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 55, normalized size = 2.04 \[ \frac {-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 99, normalized size = 3.67 \[ -\frac {2 \, {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{3 \, {\left (a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 53, normalized size = 1.96 \[ \frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-3\right )}{3\,a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.15, size = 153, normalized size = 5.67 \[ \begin {cases} - \frac {6 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {2}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{2}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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