Optimal. Leaf size=184 \[ -\frac{118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}+\frac{146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac{268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac{130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5} \]
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Rubi [A] time = 0.409626, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2765, 2977, 2968, 3019, 2750, 2648} \[ -\frac{118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}+\frac{146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac{268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac{130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2968
Rule 3019
Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{\int \frac{\cos ^3(c+d x) (4 a-10 a \cos (c+d x))}{(a+a \cos (c+d x))^5} \, dx}{11 a^2}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac{\int \frac{\cos ^2(c+d x) \left (42 a^2-76 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx}{99 a^4}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac{118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos (c+d x) \left (236 a^3-414 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{693 a^6}\\ &=-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac{118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac{\int \frac{236 a^3 \cos (c+d x)-414 a^3 \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{693 a^6}\\ &=\frac{130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac{118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}+\frac{\int \frac{-1950 a^4+2070 a^4 \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{3465 a^8}\\ &=\frac{130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac{118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac{268 \sin (c+d x)}{693 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac{146 \int \frac{1}{a+a \cos (c+d x)} \, dx}{693 a^5}\\ &=\frac{130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac{\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac{118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac{268 \sin (c+d x)}{693 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac{146 \sin (c+d x)}{693 d \left (a^6+a^6 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.343251, size = 164, normalized size = 0.89 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-33726 \sin \left (c+\frac{d x}{2}\right )+25080 \sin \left (c+\frac{3 d x}{2}\right )-23100 \sin \left (2 c+\frac{3 d x}{2}\right )+12540 \sin \left (2 c+\frac{5 d x}{2}\right )-11550 \sin \left (3 c+\frac{5 d x}{2}\right )+4565 \sin \left (3 c+\frac{7 d x}{2}\right )-3465 \sin \left (4 c+\frac{7 d x}{2}\right )+913 \sin \left (4 c+\frac{9 d x}{2}\right )-693 \sin \left (5 c+\frac{9 d x}{2}\right )+146 \sin \left (5 c+\frac{11 d x}{2}\right )+33726 \sin \left (\frac{d x}{2}\right )\right ) \sec ^{11}\left (\frac{1}{2} (c+d x)\right )}{709632 a^6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 84, normalized size = 0.5 \begin{align*}{\frac{1}{32\,d{a}^{6}} \left ( -{\frac{1}{11} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}+{\frac{5}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{10}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-{\frac{5}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13969, size = 171, normalized size = 0.93 \begin{align*} \frac{\frac{693 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1386 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{22176 \, a^{6} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53696, size = 382, normalized size = 2.08 \begin{align*} \frac{{\left (146 \, \cos \left (d x + c\right )^{5} + 183 \, \cos \left (d x + c\right )^{4} + 184 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 48 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{693 \,{\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35491, size = 115, normalized size = 0.62 \begin{align*} -\frac{63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 990 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1386 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1155 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 693 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{22176 \, a^{6} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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