Optimal. Leaf size=245 \[ -\frac{8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A-8 B+21 C)}{2 a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.691098, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {3041, 2977, 2734} \[ -\frac{8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A-8 B+21 C)}{2 a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3041
Rule 2977
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A-B+C) \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) (a (2 A+5 B-5 C)+a (2 A-2 B+9 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(B-2 C) \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 (28 a^2 (B-2 C)+a^2 (10 A-24 B+73 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(10 A-52 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(B-2 C) \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 (-3 a^3 (10 A-52 B+129 C)+a^3 (50 A-176 B+477 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(10 A-52 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(B-2 C) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{4 (20 A-83 B+216 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \cos (c+d x) \left (-8 a^4 (20 A-83 B+216 C)+105 a^4 (2 A-8 B+21 C) \cos (c+d x)\right ) \, dx}{105 a^8}\\ &=\frac{(2 A-8 B+21 C) x}{2 a^4}-\frac{8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}+\frac{(2 A-8 B+21 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac{(10 A-52 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(B-2 C) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{4 (20 A-83 B+216 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 2.79694, size = 299, normalized size = 1.22 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (210 \cos ^7\left (\frac{1}{2} (c+d x)\right ) (2 d x (2 A-8 B+21 C)+4 (B-4 C) \sin (c+d x)+C \sin (2 (c+d x)))+4 \tan \left (\frac{c}{2}\right ) (160 A-286 B+447 C) \cos ^5\left (\frac{1}{2} (c+d x)\right )-6 \tan \left (\frac{c}{2}\right ) (25 A-32 B+39 C) \cos ^3\left (\frac{1}{2} (c+d x)\right )+15 \tan \left (\frac{c}{2}\right ) (A-B+C) \cos \left (\frac{1}{2} (c+d x)\right )+15 \sec \left (\frac{c}{2}\right ) (A-B+C) \sin \left (\frac{d x}{2}\right )-8 \sec \left (\frac{c}{2}\right ) (260 A-764 B+1653 C) \sin \left (\frac{d x}{2}\right ) \cos ^6\left (\frac{1}{2} (c+d x)\right )+4 \sec \left (\frac{c}{2}\right ) (160 A-286 B+447 C) \sin \left (\frac{d x}{2}\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right )-6 \sec \left (\frac{c}{2}\right ) (25 A-32 B+39 C) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{105 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 429, normalized size = 1.8 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{B}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{C}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{9\,C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{11\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{23\,B}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{13\,C}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{15\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{49\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{111\,C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}B}{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-9\,{\frac{C \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-7\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{d{a}^{4}}}-8\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{d{a}^{4}}}+21\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.57059, size = 640, normalized size = 2.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11227, size = 730, normalized size = 2.98 \begin{align*} \frac{105 \,{\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \,{\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \,{\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \,{\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right ) + 105 \,{\left (2 \, A - 8 \, B + 21 \, C\right )} d x +{\left (105 \, C \cos \left (d x + c\right )^{5} + 210 \,{\left (B - 2 \, C\right )} \cos \left (d x + c\right )^{4} - 4 \,{\left (130 \, A - 592 \, B + 1509 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \,{\left (310 \, A - 1318 \, B + 3411 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (1070 \, A - 4472 \, B + 11619 \, C\right )} \cos \left (d x + c\right ) - 320 \, A + 1328 \, B - 3456 \, C\right )} \sin \left (d x + c\right )}{210 \,{\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 [A] time = 99.8879, size = 1624, normalized size = 6.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25714, size = 408, normalized size = 1.67 \begin{align*} \frac{\frac{420 \,{\left (d x + c\right )}{\left (2 \, A - 8 \, B + 21 \, C\right )}}{a^{4}} + \frac{840 \,{\left (2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 7 \, C \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{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 147 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 189 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1365 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5145 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 11655 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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