Optimal. Leaf size=22 \[ -\frac{1}{2 b d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 0.0263888, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2668, 32} \[ -\frac{1}{2 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{1}{2 b d (a+b \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.0244853, size = 22, normalized size = 1. \[ -\frac{1}{2 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,bd \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941628, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{2} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.4696, size = 96, normalized size = 4.36 \begin{align*} \frac{1}{2 \,{\left (b^{3} d \cos \left (d x + c\right )^{2} - 2 \, a b^{2} d \sin \left (d x + c\right ) -{\left (a^{2} b + b^{3}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.03525, size = 73, normalized size = 3.32 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{3}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{3} d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{3}} & \text{for}\: d = 0 \\- \frac{1}{2 a^{2} b d + 4 a b^{2} d \sin{\left (c + d x \right )} + 2 b^{3} d \sin ^{2}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10792, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{2} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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