Optimal. Leaf size=184 \[ -\frac{576 \sin (c+d x)}{35 a^4 d}-\frac{43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{21 x}{2 a^4}-\frac{\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.381852, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2765, 2977, 2734} \[ -\frac{576 \sin (c+d x)}{35 a^4 d}-\frac{43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{21 x}{2 a^4}-\frac{\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos ^4(c+d x) (5 a-9 a \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos ^3(c+d x) \left (56 a^2-73 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac{\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos ^2(c+d x) \left (387 a^3-477 a^3 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac{\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{288 \cos ^2(c+d x) \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{\int \cos (c+d x) \left (1728 a^4-2205 a^4 \cos (c+d x)\right ) \, dx}{105 a^8}\\ &=\frac{21 x}{2 a^4}-\frac{576 \sin (c+d x)}{35 a^4 d}+\frac{21 \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac{43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac{\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{288 \cos ^2(c+d x) \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.539259, size = 289, normalized size = 1.57 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (128730 \sin \left (c+\frac{d x}{2}\right )-140826 \sin \left (c+\frac{3 d x}{2}\right )+44310 \sin \left (2 c+\frac{3 d x}{2}\right )-60487 \sin \left (2 c+\frac{5 d x}{2}\right )+1225 \sin \left (3 c+\frac{5 d x}{2}\right )-12001 \sin \left (3 c+\frac{7 d x}{2}\right )-3185 \sin \left (4 c+\frac{7 d x}{2}\right )-315 \sin \left (4 c+\frac{9 d x}{2}\right )-315 \sin \left (5 c+\frac{9 d x}{2}\right )+35 \sin \left (5 c+\frac{11 d x}{2}\right )+35 \sin \left (6 c+\frac{11 d x}{2}\right )+102900 d x \cos \left (c+\frac{d x}{2}\right )+61740 d x \cos \left (c+\frac{3 d x}{2}\right )+61740 d x \cos \left (2 c+\frac{3 d x}{2}\right )+20580 d x \cos \left (2 c+\frac{5 d x}{2}\right )+20580 d x \cos \left (3 c+\frac{5 d x}{2}\right )+2940 d x \cos \left (3 c+\frac{7 d x}{2}\right )+2940 d x \cos \left (4 c+\frac{7 d x}{2}\right )-179830 \sin \left (\frac{d x}{2}\right )+102900 d x \cos \left (\frac{d x}{2}\right )\right )}{35840 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 160, normalized size = 0.9 \begin{align*}{\frac{1}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{9}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{13}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{111}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-9\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-7\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+21\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75057, size = 275, normalized size = 1.49 \begin{align*} -\frac{\frac{280 \,{\left (\frac{7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac{2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{63 \, \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}}}{a^{4}} - \frac{5880 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76336, size = 470, normalized size = 2.55 \begin{align*} \frac{735 \, d x \cos \left (d x + c\right )^{4} + 2940 \, d x \cos \left (d x + c\right )^{3} + 4410 \, d x \cos \left (d x + c\right )^{2} + 2940 \, d x \cos \left (d x + c\right ) + 735 \, d x +{\left (35 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 2012 \, \cos \left (d x + c\right )^{3} - 4548 \, \cos \left (d x + c\right )^{2} - 3873 \, \cos \left (d x + c\right ) - 1152\right )} \sin \left (d x + c\right )}{70 \,{\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 )}} \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 = 1.27903, size = 173, normalized size = 0.94 \begin{align*} \frac{\frac{2940 \,{\left (d x + c\right )}}{a^{4}} - \frac{280 \,{\left (9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac{5 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 63 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 455 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3885 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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