Optimal. Leaf size=73 \[ \frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \]
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Rubi [A] time = 0.0764606, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2667, 43} \[ \frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{7/2}-4 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 (a+a \sin (c+d x))^{9/2}}{9 a^3 d}-\frac{8 (a+a \sin (c+d x))^{11/2}}{11 a^4 d}+\frac{2 (a+a \sin (c+d x))^{13/2}}{13 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.158593, size = 51, normalized size = 0.7 \[ \frac{2 (\sin (c+d x)+1)^3 \left (99 \sin ^2(c+d x)-270 \sin (c+d x)+203\right ) (a (\sin (c+d x)+1))^{3/2}}{1287 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 41, normalized size = 0.6 \begin{align*} -{\frac{198\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+540\,\sin \left ( dx+c \right ) -604}{1287\,{a}^{3}d} \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 = 0.96499, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (99 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 468 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 572 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2}\right )}}{1287 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70304, size = 242, normalized size = 3.32 \begin{align*} -\frac{2 \,{\left (99 \, a \cos \left (d x + c\right )^{6} - 14 \, a \cos \left (d x + c\right )^{4} - 32 \, a \cos \left (d x + c\right )^{2} - 2 \,{\left (63 \, a \cos \left (d x + c\right )^{4} + 80 \, a \cos \left (d x + c\right )^{2} + 128 \, a\right )} \sin \left (d x + c\right ) - 256 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{1287 \, 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{3}{2}} \cos \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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