Optimal. Leaf size=27 \[ -\frac {\cos ^3(c+d x)}{3 d (a+a \sin (c+d x))^3} \]
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Rubi [A]
time = 0.03, 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 = {2750}
\begin {gather*} -\frac {\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
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 = 0.01, size = 28, normalized size = 1.04 \begin {gather*} -\frac {\cos ^3(c+d x)}{3 a^3 d (1+\sin (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs.
\(2(25)=50\).
time = 0.21, size = 55, normalized size = 2.04
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}-\frac {2}{3}}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3}}\) | \(36\) |
derivativedivides | \(\frac {-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{d \,a^{3}}\) | \(55\) |
default | \(\frac {-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{d \,a^{3}}\) | \(55\) |
norman | \(\frac {-\frac {2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2}{3 a d}-\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {20 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {28 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (25) = 50\).
time = 0.28, size = 99, normalized size = 3.67 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (25) = 50\).
time = 0.35, size = 95, normalized size = 3.52 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs.
\(2 (24) = 48\).
time = 5.09, size = 153, normalized size = 5.67 \begin {gather*} \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}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.90, size = 36, normalized size = 1.33 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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