Optimal. Leaf size=47 \[ \frac{(a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (a-a \sin (c+d x))^3}{3 a^4 d} \]
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Rubi [A] time = 0.0562803, 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} \[ \frac{(a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (a-a \sin (c+d x))^3}{3 a^4 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a-x)^2-(a-x)^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac{2 (a-a \sin (c+d x))^3}{3 a^4 d}+\frac{(a-a \sin (c+d x))^4}{4 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.0932865, size = 46, normalized size = 0.98 \[ \frac{\sin (c+d x) \left (3 \sin ^3(c+d x)-4 \sin ^2(c+d x)-6 \sin (c+d x)+12\right )}{12 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 45, normalized size = 1. \begin{align*}{\frac{1}{da} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{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 = 0.930849, size = 63, normalized size = 1.34 \begin{align*} \frac{3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} + 12 \, \sin \left (d x + c\right )}{12 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67223, size = 93, normalized size = 1.98 \begin{align*} \frac{3 \, \cos \left (d x + c\right )^{4} + 4 \,{\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{12 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 126.403, size = 779, normalized size = 16.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14575, size = 63, normalized size = 1.34 \begin{align*} \frac{3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} + 12 \, \sin \left (d x + c\right )}{12 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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