Optimal. Leaf size=23 \[ -\frac{(a-a \sin (c+d x))^4}{4 a^7 d} \]
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Rubi [A] time = 0.0428404, 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} \[ -\frac{(a-a \sin (c+d x))^4}{4 a^7 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac{(a-a \sin (c+d x))^4}{4 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.158463, size = 44, normalized size = 1.91 \[ -\frac{\sin (c+d x) \left (\sin ^3(c+d x)-4 \sin ^2(c+d x)+6 \sin (c+d x)-4\right )}{4 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 19, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}{4\,d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.970327, size = 61, normalized size = 2.65 \begin{align*} -\frac{\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{4 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9082, size = 119, normalized size = 5.17 \begin{align*} -\frac{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \,{\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right )}{4 \, a^{3} 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 [B] time = 1.17269, size = 61, normalized size = 2.65 \begin{align*} -\frac{\sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{4 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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