Optimal. Leaf size=47 \[ \frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d} \]
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Rubi [A] time = 0.0520617, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a-x)^3-(a-x)^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac{(a-a \sin (c+d x))^4}{2 a^6 d}+\frac{(a-a \sin (c+d x))^5}{5 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.159816, size = 46, normalized size = 0.98 \[ -\frac{\sin (c+d x) \left (2 \sin ^4(c+d x)-5 \sin ^3(c+d x)+10 \sin (c+d x)-10\right )}{10 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 45, normalized size = 1. \begin{align*}{\frac{1}{d{a}^{2}} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2}}- \left ( \sin \left ( dx+c \right ) \right ) ^{2}+\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937682, size = 63, normalized size = 1.34 \begin{align*} -\frac{2 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{4} + 10 \, \sin \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right )}{10 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93265, size = 122, normalized size = 2.6 \begin{align*} \frac{5 \, \cos \left (d x + c\right )^{4} - 2 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right )}{10 \, 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 = 1.14583, size = 63, normalized size = 1.34 \begin{align*} -\frac{2 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{4} + 10 \, \sin \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right )}{10 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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