Optimal. Leaf size=99 \[ -\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}-\frac{(A-2 B) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (A-2 B)}{a^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.275746, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2977, 2968, 3023, 12, 2735, 2648} \[ -\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}-\frac{(A-2 B) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (A-2 B)}{a^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2977
Rule 2968
Rule 3023
Rule 12
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx &=\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\cos (c+d x) (2 a (A-B)-a (A-4 B) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{2 a (A-B) \cos (c+d x)-a (A-4 B) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}+\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{3 a^2 (A-2 B) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^3}\\ &=-\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}+\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(A-2 B) \int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac{(A-2 B) x}{a^2}-\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}+\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(A-2 B) \int \frac{1}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac{(A-2 B) x}{a^2}-\frac{(A-4 B) \sin (c+d x)}{3 a^2 d}+\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(A-2 B) \sin (c+d x)}{d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.677156, size = 137, normalized size = 1.38 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (6 \cos ^3\left (\frac{1}{2} (c+d x)\right ) (d x (A-2 B)+B \sin (c+d x))+(A-B) \tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+(A-B) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )-2 (5 A-8 B) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 149, normalized size = 1.5 \begin{align*}{\frac{A}{6\,{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{B}{6\,{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{3\,A}{2\,{a}^{2}d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{5\,B}{2\,{a}^{2}d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{{a}^{2}d \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{{a}^{2}d}}-4\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.89457, size = 258, normalized size = 2.61 \begin{align*} \frac{B{\left (\frac{\frac{15 \, \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{24 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac{12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac{a^{2} \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 )}}\right )} - A{\left (\frac{\frac{9 \, \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{12 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36591, size = 294, normalized size = 2.97 \begin{align*} \frac{3 \,{\left (A - 2 \, B\right )} d x \cos \left (d x + c\right )^{2} + 6 \,{\left (A - 2 \, B\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (A - 2 \, B\right )} d x +{\left (3 \, B \cos \left (d x + c\right )^{2} -{\left (5 \, A - 14 \, B\right )} \cos \left (d x + c\right ) - 4 \, A + 10 \, B\right )} \sin \left (d x + c\right )}{3 \,{\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 = 7.01014, size = 411, normalized size = 4.15 \begin{align*} \begin{cases} \frac{6 A d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{6 A d x}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{A \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{8 A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{9 A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{12 B d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{12 B d x}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{B \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{14 B \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{27 B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + B \cos{\left (c \right )}\right ) \cos ^{2}{\left (c \right )}}{\left (a \cos{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26666, size = 161, normalized size = 1.63 \begin{align*} \frac{\frac{6 \,{\left (d x + c\right )}{\left (A - 2 \, B\right )}}{a^{2}} + \frac{12 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{2}} + \frac{A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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