Optimal. Leaf size=216 \[ -\frac{4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(6 A+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A+23 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.484895, antiderivative size = 216, 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, 2635, 8, 2633} \[ -\frac{4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(6 A+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A+23 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2977
Rule 2748
Rule 2635
Rule 8
Rule 2633
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))^3} \, dx &=-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos ^4(c+d x) (-5 a C+a (3 A+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(3 A+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{\int \frac{\cos ^3(c+d x) \left (-4 a^2 (3 A+13 C)+9 a^2 (2 A+7 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(3 A+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(6 A+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int \cos ^2(c+d x) \left (-15 a^3 (6 A+23 C)+12 a^3 (9 A+34 C) \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(3 A+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(6 A+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{(6 A+23 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac{(4 (9 A+34 C)) \int \cos ^3(c+d x) \, dx}{5 a^3}\\ &=-\frac{(6 A+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(3 A+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(6 A+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{(6 A+23 C) \int 1 \, dx}{2 a^3}-\frac{(4 (9 A+34 C)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac{(6 A+23 C) x}{2 a^3}+\frac{4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(3 A+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(6 A+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d}\\ \end{align*}
Mathematica [B] time = 0.852709, size = 463, normalized size = 2.14 \[ -\frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (600 d x (6 A+23 C) \cos \left (c+\frac{d x}{2}\right )+4500 A \sin \left (c+\frac{d x}{2}\right )-4860 A \sin \left (c+\frac{3 d x}{2}\right )+900 A \sin \left (2 c+\frac{3 d x}{2}\right )-1452 A \sin \left (2 c+\frac{5 d x}{2}\right )-300 A \sin \left (3 c+\frac{5 d x}{2}\right )-60 A \sin \left (3 c+\frac{7 d x}{2}\right )-60 A \sin \left (4 c+\frac{7 d x}{2}\right )+1800 A d x \cos \left (c+\frac{3 d x}{2}\right )+1800 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+360 A d x \cos \left (2 c+\frac{5 d x}{2}\right )+360 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+600 d x (6 A+23 C) \cos \left (\frac{d x}{2}\right )-7020 A \sin \left (\frac{d x}{2}\right )+11110 C \sin \left (c+\frac{d x}{2}\right )-15380 C \sin \left (c+\frac{3 d x}{2}\right )+380 C \sin \left (2 c+\frac{3 d x}{2}\right )-4777 C \sin \left (2 c+\frac{5 d x}{2}\right )-1625 C \sin \left (3 c+\frac{5 d x}{2}\right )-230 C \sin \left (3 c+\frac{7 d x}{2}\right )-230 C \sin \left (4 c+\frac{7 d x}{2}\right )+20 C \sin \left (4 c+\frac{9 d x}{2}\right )+20 C \sin \left (5 c+\frac{9 d x}{2}\right )-5 C \sin \left (5 c+\frac{11 d x}{2}\right )-5 C \sin \left (6 c+\frac{11 d x}{2}\right )+6900 C d x \cos \left (c+\frac{3 d x}{2}\right )+6900 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+1380 C d x \cos \left (2 c+\frac{5 d x}{2}\right )+1380 C d x \cos \left (3 c+\frac{5 d x}{2}\right )-20410 C \sin \left (\frac{d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 362, normalized size = 1.7 \begin{align*}{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{C}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{5\,C}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{17\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{49\,C}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{A \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{d{a}^{3} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+17\,{\frac{C \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{d{a}^{3} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}A}{d{a}^{3} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+{\frac{76\,C}{3\,d{a}^{3}} \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 ) ^{-3}}+2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+11\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-6\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{d{a}^{3}}}-23\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59266, size = 493, normalized size = 2.28 \begin{align*} \frac{C{\left (\frac{20 \,{\left (\frac{33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\frac{735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{1380 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, A{\left (\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac{a^{3} \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{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74121, size = 525, normalized size = 2.43 \begin{align*} -\frac{15 \,{\left (6 \, A + 23 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \,{\left (6 \, A + 23 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \,{\left (6 \, A + 23 \, C\right )} d x \cos \left (d x + c\right ) + 15 \,{\left (6 \, A + 23 \, C\right )} d x -{\left (10 \, C \cos \left (d x + c\right )^{5} - 15 \, C \cos \left (d x + c\right )^{4} + 5 \,{\left (6 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (234 \, A + 869 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \,{\left (38 \, A + 143 \, C\right )} \cos \left (d x + c\right ) + 144 \, A + 544 \, C\right )} \sin \left (d x + c\right )}{30 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 55.9321, size = 1586, normalized size = 7.34 \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 = 1.23347, size = 308, normalized size = 1.43 \begin{align*} -\frac{\frac{30 \,{\left (d x + c\right )}{\left (6 \, A + 23 \, C\right )}}{a^{3}} - \frac{20 \,{\left (6 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 51 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 76 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 33 \, 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 )}^{3} a^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 50 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 735 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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