Optimal. Leaf size=112 \[ \frac{8 \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{8 \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{4 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.0783698, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2750, 2650, 2648} \[ \frac{8 \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{8 \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{4 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{4 \int \frac{1}{(a+a \cos (c+d x))^3} \, dx}{7 a}\\ &=-\frac{\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{8 \int \frac{1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2}\\ &=-\frac{\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{8 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{8 \int \frac{1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=-\frac{\sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{4 \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{8 \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{8 \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.220062, size = 87, normalized size = 0.78 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-35 \sin \left (c+\frac{d x}{2}\right )+2 \left (21 \sin \left (c+\frac{3 d x}{2}\right )+7 \sin \left (2 c+\frac{5 d x}{2}\right )+\sin \left (3 c+\frac{7 d x}{2}\right )\right )+35 \sin \left (\frac{d x}{2}\right )\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right )}{1680 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 58, normalized size = 0.5 \begin{align*}{\frac{1}{8\,d{a}^{4}} \left ( -{\frac{1}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{1}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+\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 [A] time = 1.16435, size = 117, normalized size = 1.04 \begin{align*} \frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{35 \, \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{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{840 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53257, size = 251, normalized size = 2.24 \begin{align*} \frac{{\left (8 \, \cos \left (d x + c\right )^{3} + 32 \, \cos \left (d x + c\right )^{2} + 52 \, \cos \left (d x + c\right ) + 13\right )} \sin \left (d x + c\right )}{105 \,{\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 = 7.78498, size = 85, normalized size = 0.76 \begin{align*} \begin{cases} - \frac{\tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{56 a^{4} d} - \frac{\tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{40 a^{4} d} + \frac{\tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{24 a^{4} d} + \frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{\left (a \cos{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35611, size = 80, normalized size = 0.71 \begin{align*} -\frac{15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{840 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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