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 = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.13, 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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fricas [B] time = 0.75, size = 56, normalized size = 2.15 \[ -\frac {B \sin \left (d x + c\right )}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 47, normalized size = 1.81 \[ -\frac {B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{8 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 48, normalized size = 1.85 \[ \frac {B \left (-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 115, normalized size = 4.42 \[ -\frac {\frac {B {\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {2 \, B {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 33, normalized size = 1.27 \[ -\frac {B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}{8\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.37, size = 80, normalized size = 3.08 \[ \begin {cases} - \frac {B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{3} d} - \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4 a^{3} d} - \frac {B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \left (B \cos {\relax (c )} - \frac {3 B}{2}\right )}{\left (a \cos {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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