Optimal. Leaf size=223 \[ -\frac{32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{16 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A+21 C)}{2 a^4}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.612062, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3042, 2977, 2734} \[ -\frac{32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{16 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A+21 C)}{2 a^4}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2977
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A+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 C)+a (2 A+9 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{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 (-56 a^2 C+a^2 (10 A+73 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(10 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{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+129 C)+a^3 (50 A+477 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(10 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{16 (5 A+54 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 (-32 a^4 (5 A+54 C)+105 a^4 (2 A+21 C) \cos (c+d x)\right ) \, dx}{105 a^8}\\ &=\frac{(2 A+21 C) x}{2 a^4}-\frac{32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}+\frac{(2 A+21 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac{(10 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac{16 (5 A+54 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 [B] time = 1.04144, size = 513, normalized size = 2.3 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (14700 d x (2 A+21 C) \cos \left (c+\frac{d x}{2}\right )+66080 A \sin \left (c+\frac{d x}{2}\right )-57120 A \sin \left (c+\frac{3 d x}{2}\right )+30240 A \sin \left (2 c+\frac{3 d x}{2}\right )-22400 A \sin \left (2 c+\frac{5 d x}{2}\right )+6720 A \sin \left (3 c+\frac{5 d x}{2}\right )-4160 A \sin \left (3 c+\frac{7 d x}{2}\right )+17640 A d x \cos \left (c+\frac{3 d x}{2}\right )+17640 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+5880 A d x \cos \left (2 c+\frac{5 d x}{2}\right )+5880 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+840 A d x \cos \left (3 c+\frac{7 d x}{2}\right )+840 A d x \cos \left (4 c+\frac{7 d x}{2}\right )+14700 d x (2 A+21 C) \cos \left (\frac{d x}{2}\right )-79520 A \sin \left (\frac{d x}{2}\right )+386190 C \sin \left (c+\frac{d x}{2}\right )-422478 C \sin \left (c+\frac{3 d x}{2}\right )+132930 C \sin \left (2 c+\frac{3 d x}{2}\right )-181461 C \sin \left (2 c+\frac{5 d x}{2}\right )+3675 C \sin \left (3 c+\frac{5 d x}{2}\right )-36003 C \sin \left (3 c+\frac{7 d x}{2}\right )-9555 C \sin \left (4 c+\frac{7 d x}{2}\right )-945 C \sin \left (4 c+\frac{9 d x}{2}\right )-945 C \sin \left (5 c+\frac{9 d x}{2}\right )+105 C \sin \left (5 c+\frac{11 d x}{2}\right )+105 C \sin \left (6 c+\frac{11 d x}{2}\right )+185220 C d x \cos \left (c+\frac{3 d x}{2}\right )+185220 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+61740 C d x \cos \left (2 c+\frac{5 d x}{2}\right )+61740 C d x \cos \left (3 c+\frac{5 d x}{2}\right )+8820 C d x \cos \left (3 c+\frac{7 d x}{2}\right )+8820 C d x \cos \left (4 c+\frac{7 d x}{2}\right )-539490 C \sin \left (\frac{d x}{2}\right )\right )}{6720 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 264, normalized size = 1.2 \begin{align*}{\frac{A}{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{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{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{111\,C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-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}}}-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}}}+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 [A] time = 1.60033, size = 429, normalized size = 1.92 \begin{align*} -\frac{3 \, C{\left (\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}}\right )} + 5 \, A{\left (\frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{77 \, \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{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{336 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51275, size = 632, normalized size = 2.83 \begin{align*} \frac{105 \,{\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \,{\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \,{\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \,{\left (2 \, A + 21 \, C\right )} d x \cos \left (d x + c\right ) + 105 \,{\left (2 \, A + 21 \, C\right )} d x +{\left (105 \, C \cos \left (d x + c\right )^{5} - 420 \, C \cos \left (d x + c\right )^{4} - 4 \,{\left (130 \, A + 1509 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \,{\left (310 \, A + 3411 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (1070 \, A + 11619 \, C\right )} \cos \left (d x + c\right ) - 320 \, A - 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 = 71.1258, size = 1086, normalized size = 4.87 \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 = 1.24915, size = 279, normalized size = 1.25 \begin{align*} \frac{\frac{420 \,{\left (d x + c\right )}{\left (2 \, A + 21 \, C\right )}}{a^{4}} - \frac{840 \,{\left (9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 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 \, 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} - 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} + 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 ) - 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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