Optimal. Leaf size=114 \[ \frac {12 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac {18 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {8 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.20, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2765, 2968, 3019, 2750, 2648} \[ \frac {12 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac {18 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {8 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2750
Rule 2765
Rule 2968
Rule 3019
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {\cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos (c+d x) (2 a-6 a \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {\int \frac {2 a \cos (c+d x)-6 a \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {8 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {-24 a^2+30 a^2 \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {18 \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {8 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {12 \int \frac {1}{a+a \cos (c+d x)} \, dx}{35 a^3}\\ &=-\frac {18 \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {8 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {12 \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 112, normalized size = 0.98 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-210 \sin \left (c+\frac {d x}{2}\right )+147 \sin \left (c+\frac {3 d x}{2}\right )-105 \sin \left (2 c+\frac {3 d x}{2}\right )+49 \sin \left (2 c+\frac {5 d x}{2}\right )-35 \sin \left (3 c+\frac {5 d x}{2}\right )+12 \sin \left (3 c+\frac {7 d x}{2}\right )+210 \sin \left (\frac {d x}{2}\right )\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right )}{2240 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 99, normalized size = 0.87 \[ \frac {{\left (12 \, \cos \left (d x + c\right )^{3} + 13 \, \cos \left (d x + c\right )^{2} + 8 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{35 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 59, normalized size = 0.52 \[ -\frac {5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{280 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 58, normalized size = 0.51 \[ \frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 87, normalized size = 0.76 \[ \frac {\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{280 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 58, normalized size = 0.51 \[ -\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-35\right )}{280\,a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.35, size = 88, normalized size = 0.77 \[ \begin {cases} - \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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