Optimal. Leaf size=191 \[ \frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac{2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac{(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{x (28 A+55 C)}{8 a^2}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.3378, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3042, 2977, 2748, 2633, 2635, 8} \[ \frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac{2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac{(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{x (28 A+55 C)}{8 a^2}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2977
Rule 2748
Rule 2633
Rule 2635
Rule 8
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))^2} \, dx &=-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\cos ^4(c+d x) (-a (2 A+5 C)+a (4 A+7 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \cos ^3(c+d x) \left (-24 a^2 (A+2 C)+a^2 (28 A+55 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(8 (A+2 C)) \int \cos ^3(c+d x) \, dx}{a^2}+\frac{(28 A+55 C) \int \cos ^4(c+d x) \, dx}{3 a^2}\\ &=\frac{(28 A+55 C) \cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac{2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(28 A+55 C) \int \cos ^2(c+d x) \, dx}{4 a^2}+\frac{(8 (A+2 C)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}+\frac{(28 A+55 C) \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{(28 A+55 C) \cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac{2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(28 A+55 C) \int 1 \, dx}{8 a^2}\\ &=\frac{(28 A+55 C) x}{8 a^2}-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}+\frac{(28 A+55 C) \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{(28 A+55 C) \cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac{2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [B] time = 0.779352, size = 399, normalized size = 2.09 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (72 d x (28 A+55 C) \cos \left (c+\frac{d x}{2}\right )+1176 A \sin \left (c+\frac{d x}{2}\right )-1912 A \sin \left (c+\frac{3 d x}{2}\right )-504 A \sin \left (2 c+\frac{3 d x}{2}\right )-120 A \sin \left (2 c+\frac{5 d x}{2}\right )-120 A \sin \left (3 c+\frac{5 d x}{2}\right )+24 A \sin \left (3 c+\frac{7 d x}{2}\right )+24 A \sin \left (4 c+\frac{7 d x}{2}\right )+672 A d x \cos \left (c+\frac{3 d x}{2}\right )+672 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+72 d x (28 A+55 C) \cos \left (\frac{d x}{2}\right )-3048 A \sin \left (\frac{d x}{2}\right )+1344 C \sin \left (c+\frac{d x}{2}\right )-3488 C \sin \left (c+\frac{3 d x}{2}\right )-1312 C \sin \left (2 c+\frac{3 d x}{2}\right )-285 C \sin \left (2 c+\frac{5 d x}{2}\right )-285 C \sin \left (3 c+\frac{5 d x}{2}\right )+57 C \sin \left (3 c+\frac{7 d x}{2}\right )+57 C \sin \left (4 c+\frac{7 d x}{2}\right )-7 C \sin \left (4 c+\frac{9 d x}{2}\right )-7 C \sin \left (5 c+\frac{9 d x}{2}\right )+3 C \sin \left (5 c+\frac{11 d x}{2}\right )+3 C \sin \left (6 c+\frac{11 d x}{2}\right )+1320 C d x \cos \left (c+\frac{3 d x}{2}\right )+1320 C d x \cos \left (2 c+\frac{3 d x}{2}\right )-5184 C \sin \left (\frac{d x}{2}\right )\right )}{384 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.034, size = 392, normalized size = 2.1 \begin{align*}{\frac{A}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{C}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{7\,A}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{11\,C}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}A}{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{4}}}-{\frac{65\,C}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-13\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}A}{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{4}}}-{\frac{395\,C}{12\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-11\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}A}{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{4}}}-{\frac{341\,C}{12\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-3\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{4}}}-{\frac{31\,C}{4\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+7\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{d{a}^{2}}}+{\frac{55\,C}{4\,d{a}^{2}}\arctan \left ( \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 [B] time = 1.55908, size = 560, normalized size = 2.93 \begin{align*} -\frac{C{\left (\frac{\frac{93 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{341 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{195 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac{2 \,{\left (\frac{33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac{165 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} + 2 \, A{\left (\frac{6 \,{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac{2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{42 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41821, size = 432, normalized size = 2.26 \begin{align*} \frac{3 \,{\left (28 \, A + 55 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \,{\left (28 \, A + 55 \, C\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (28 \, A + 55 \, C\right )} d x +{\left (6 \, C \cos \left (d x + c\right )^{5} - 4 \, C \cos \left (d x + c\right )^{4} +{\left (12 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{3} - 6 \,{\left (4 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (172 \, A + 347 \, C\right )} \cos \left (d x + c\right ) - 128 \, A - 256 \, C\right )} \sin \left (d x + c\right )}{24 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.6805, size = 2161, normalized size = 11.31 \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 = 1.18072, size = 297, normalized size = 1.55 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}{\left (28 \, A + 55 \, C\right )}}{a^{2}} + \frac{4 \,{\left (A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 21 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 33 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{6}} - \frac{2 \,{\left (60 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 195 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 156 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 395 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 132 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 341 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 36 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 93 \, 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 )}^{4} a^{2}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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