Optimal. Leaf size=191 \[ \frac{181 \sin (c+d x)}{63 a^5 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}+\frac{5 \sin (c+d x)}{d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{5 x}{a^5}-\frac{\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.490209, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2765, 2977, 2968, 3023, 12, 2735, 2648} \[ \frac{181 \sin (c+d x)}{63 a^5 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}+\frac{5 \sin (c+d x)}{d \left (a^5 \cos (c+d x)+a^5\right )}-\frac{5 x}{a^5}-\frac{\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2968
Rule 3023
Rule 12
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{\int \frac{\cos ^4(c+d x) (5 a-10 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos ^3(c+d x) \left (60 a^2-85 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos ^2(c+d x) \left (435 a^3-570 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{\int \frac{\cos (c+d x) \left (2010 a^4-2715 a^4 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{\int \frac{2010 a^4 \cos (c+d x)-2715 a^4 \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=\frac{181 \sin (c+d x)}{63 a^5 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{\int \frac{4725 a^5 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^9}\\ &=\frac{181 \sin (c+d x)}{63 a^5 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}-\frac{5 \int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^4}\\ &=-\frac{5 x}{a^5}+\frac{181 \sin (c+d x)}{63 a^5 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac{5 \int \frac{1}{a+a \cos (c+d x)} \, dx}{a^4}\\ &=-\frac{5 x}{a^5}+\frac{181 \sin (c+d x)}{63 a^5 d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac{5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac{29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac{67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac{5 \sin (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.771424, size = 319, normalized size = 1.67 \[ -\frac{\sec \left (\frac{c}{2}\right ) \sec ^9\left (\frac{1}{2} (c+d x)\right ) \left (143010 \sin \left (c+\frac{d x}{2}\right )-138726 \sin \left (c+\frac{3 d x}{2}\right )+73290 \sin \left (2 c+\frac{3 d x}{2}\right )-70389 \sin \left (2 c+\frac{5 d x}{2}\right )+20475 \sin \left (3 c+\frac{5 d x}{2}\right )-21141 \sin \left (3 c+\frac{7 d x}{2}\right )+1575 \sin \left (4 c+\frac{7 d x}{2}\right )-3091 \sin \left (4 c+\frac{9 d x}{2}\right )-567 \sin \left (5 c+\frac{9 d x}{2}\right )-63 \sin \left (5 c+\frac{11 d x}{2}\right )-63 \sin \left (6 c+\frac{11 d x}{2}\right )+79380 d x \cos \left (c+\frac{d x}{2}\right )+52920 d x \cos \left (c+\frac{3 d x}{2}\right )+52920 d x \cos \left (2 c+\frac{3 d x}{2}\right )+22680 d x \cos \left (2 c+\frac{5 d x}{2}\right )+22680 d x \cos \left (3 c+\frac{5 d x}{2}\right )+5670 d x \cos \left (3 c+\frac{7 d x}{2}\right )+5670 d x \cos \left (4 c+\frac{7 d x}{2}\right )+630 d x \cos \left (4 c+\frac{9 d x}{2}\right )+630 d x \cos \left (5 c+\frac{9 d x}{2}\right )-175014 \sin \left (\frac{d x}{2}\right )+79380 d x \cos \left (\frac{d x}{2}\right )\right )}{64512 a^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 145, normalized size = 0.8 \begin{align*}{\frac{1}{144\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{1}{14\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3}{8\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{3}{2\,d{a}^{5}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{129}{16\,d{a}^{5}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{5} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-10\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65445, size = 240, normalized size = 1.26 \begin{align*} \frac{\frac{2016 \, \sin \left (d x + c\right )}{{\left (a^{5} + \frac{a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac{10080 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{1008 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71494, size = 541, normalized size = 2.83 \begin{align*} -\frac{315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x -{\left (63 \, \cos \left (d x + c\right )^{5} + 946 \, \cos \left (d x + c\right )^{4} + 2840 \, \cos \left (d x + c\right )^{3} + 3633 \, \cos \left (d x + c\right )^{2} + 2165 \, \cos \left (d x + c\right ) + 496\right )} \sin \left (d x + c\right )}{63 \,{\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} 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 = 1.38633, size = 174, normalized size = 0.91 \begin{align*} -\frac{\frac{5040 \,{\left (d x + c\right )}}{a^{5}} - \frac{2016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{5}} - \frac{7 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 72 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1512 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{45}}}{1008 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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