Optimal. Leaf size=23 \[ -\frac{(a-a \sin (c+d x))^3}{3 a^5 d} \]
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Rubi [A] time = 0.0431779, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 32} \[ -\frac{(a-a \sin (c+d x))^3}{3 a^5 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac{(a-a \sin (c+d x))^3}{3 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.0561295, size = 34, normalized size = 1.48 \[ \frac{\sin (c+d x) \left (\sin ^2(c+d x)-3 \sin (c+d x)+3\right )}{3 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 19, normalized size = 0.8 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}{3\,d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939798, size = 47, normalized size = 2.04 \begin{align*} \frac{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90076, size = 92, normalized size = 4. \begin{align*} \frac{3 \, \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1311, size = 47, normalized size = 2.04 \begin{align*} \frac{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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