Optimal. Leaf size=98 \[ \frac{(2 A+7 C) \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.120647, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3020, 2750, 2648} \[ \frac{(2 A+7 C) \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3020
Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{\int \frac{-a (2 A-3 C)-5 a C \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{(2 A+7 C) \int \frac{1}{a+a \cos (c+d x)} \, dx}{15 a^2}\\ &=\frac{(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{(2 A+7 C) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.27611, size = 129, normalized size = 1.32 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (10 A \sin \left (c+\frac{3 d x}{2}\right )+2 A \sin \left (2 c+\frac{5 d x}{2}\right )+20 (A+2 C) \sin \left (\frac{d x}{2}\right )-30 C \sin \left (c+\frac{d x}{2}\right )+20 C \sin \left (c+\frac{3 d x}{2}\right )-15 C \sin \left (2 c+\frac{3 d x}{2}\right )+7 C \sin \left (2 c+\frac{5 d x}{2}\right )\right )}{30 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 88, normalized size = 0.9 \begin{align*}{\frac{1}{4\,d{a}^{3}} \left ({\frac{A}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{C}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{2\,A}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{2\,C}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+A\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +C\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 = 1.02457, size = 181, normalized size = 1.85 \begin{align*} \frac{\frac{A{\left (\frac{15 \, \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{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac{C{\left (\frac{15 \, \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{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31873, size = 221, normalized size = 2.26 \begin{align*} \frac{{\left ({\left (2 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left (A + C\right )} \cos \left (d x + c\right ) + 7 \, A + 2 \, C\right )} \sin \left (d x + c\right )}{15 \,{\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 = 5.94721, size = 136, normalized size = 1.39 \begin{align*} \begin{cases} \frac{A \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{20 a^{3} d} + \frac{A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{3} d} + \frac{A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4 a^{3} d} + \frac{C \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{20 a^{3} d} - \frac{C \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{3} d} + \frac{C \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + C \cos ^{2}{\left (c \right )}\right )}{\left (a \cos{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24804, size = 120, normalized size = 1.22 \begin{align*} \frac{3 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 10 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 10 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{60 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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