Optimal. Leaf size=26 \[ -\frac{B \sin (c+d x)}{2 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.0328949, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2749} \[ -\frac{B \sin (c+d x)}{2 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2749
Rubi steps
\begin{align*} \int \frac{-\frac{3 B}{2}+B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac{B \sin (c+d x)}{2 d (a+a \cos (c+d x))^3}\\ \end{align*}
Mathematica [A] time = 0.103346, size = 27, normalized size = 1.04 \[ -\frac{B \sin (c+d x)}{2 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 48, normalized size = 1.9 \begin{align*}{\frac{B}{8\,d{a}^{3}} \left ( - \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}-2\, \left ( \tan \left ( 1/2\,dx+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 [B] time = 1.06946, size = 155, normalized size = 5.96 \begin{align*} -\frac{\frac{B{\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{2 \, B{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{40 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33963, size = 135, normalized size = 5.19 \begin{align*} -\frac{B \sin \left (d x + c\right )}{2 \,{\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 = 3.66162, size = 80, normalized size = 3.08 \begin{align*} \begin{cases} - \frac{B \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 a^{3} d} - \frac{B \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4 a^{3} d} - \frac{B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \left (B \cos{\left (c \right )} - \frac{3 B}{2}\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 = 1.32672, size = 63, normalized size = 2.42 \begin{align*} -\frac{B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{8 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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