Optimal. Leaf size=24 \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d} \]
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Rubi [A] time = 0.0332088, antiderivative size = 24, 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} \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{2 (a+a \sin (c+d x))^{9/2}}{9 a d}\\ \end{align*}
Mathematica [A] time = 0.0805537, size = 24, normalized size = 1. \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 21, normalized size = 0.9 \begin{align*}{\frac{2}{9\,da} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950317, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}}}{9 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63613, size = 176, normalized size = 7.33 \begin{align*} \frac{2 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} - 4 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{9 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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