Optimal. Leaf size=73 \[ \frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]
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Rubi [A] time = 0.0746197, 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} \[ \frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]
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))^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^{11/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{11/2}-4 a (a+x)^{13/2}+(a+x)^{15/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 (a+a \sin (c+d x))^{13/2}}{13 a^3 d}-\frac{8 (a+a \sin (c+d x))^{15/2}}{15 a^4 d}+\frac{2 (a+a \sin (c+d x))^{17/2}}{17 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.269581, size = 54, normalized size = 0.74 \[ \frac{2 a^3 (\sin (c+d x)+1)^6 \left (195 \sin ^2(c+d x)-494 \sin (c+d x)+331\right ) \sqrt{a (\sin (c+d x)+1)}}{3315 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 41, normalized size = 0.6 \begin{align*} -{\frac{390\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+988\,\sin \left ( dx+c \right ) -1052}{3315\,{a}^{3}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954958, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (195 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{17}{2}} - 884 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{15}{2}} a + 1020 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a^{2}\right )}}{3315 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67033, size = 335, normalized size = 4.59 \begin{align*} \frac{2 \,{\left (195 \, a^{3} \cos \left (d x + c\right )^{8} - 1072 \, a^{3} \cos \left (d x + c\right )^{6} + 56 \, a^{3} \cos \left (d x + c\right )^{4} + 128 \, a^{3} \cos \left (d x + c\right )^{2} + 1024 \, a^{3} - 4 \,{\left (169 \, a^{3} \cos \left (d x + c\right )^{6} - 126 \, a^{3} \cos \left (d x + c\right )^{4} - 160 \, a^{3} \cos \left (d x + c\right )^{2} - 256 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3315 \, 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 )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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