Optimal. Leaf size=97 \[ -\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d} \]
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Rubi [A] time = 0.0861733, antiderivative size = 97, 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} \[ -\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^{9/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{9/2}-12 a^2 (a+x)^{11/2}+6 a (a+x)^{13/2}-(a+x)^{15/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{16 (a+a \sin (c+d x))^{11/2}}{11 a^4 d}-\frac{24 (a+a \sin (c+d x))^{13/2}}{13 a^5 d}+\frac{4 (a+a \sin (c+d x))^{15/2}}{5 a^6 d}-\frac{2 (a+a \sin (c+d x))^{17/2}}{17 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.388605, size = 61, normalized size = 0.63 \[ -\frac{2 (\sin (c+d x)+1)^4 \left (715 \sin ^3(c+d x)-2717 \sin ^2(c+d x)+3641 \sin (c+d x)-1767\right ) (a (\sin (c+d x)+1))^{3/2}}{12155 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.096, size = 57, normalized size = 0.6 \begin{align*}{\frac{1430\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -5434\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-8712\,\sin \left ( dx+c \right ) +8968}{12155\,{a}^{4}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968551, size = 97, normalized size = 1. \begin{align*} -\frac{2 \,{\left (715 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{17}{2}} - 4862 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{15}{2}} a + 11220 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a^{2} - 8840 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a^{3}\right )}}{12155 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83001, size = 313, normalized size = 3.23 \begin{align*} -\frac{2 \,{\left (715 \, a \cos \left (d x + c\right )^{8} - 66 \, a \cos \left (d x + c\right )^{6} - 112 \, a \cos \left (d x + c\right )^{4} - 256 \, a \cos \left (d x + c\right )^{2} - 2 \,{\left (429 \, a \cos \left (d x + c\right )^{6} + 504 \, a \cos \left (d x + c\right )^{4} + 640 \, a \cos \left (d x + c\right )^{2} + 1024 \, a\right )} \sin \left (d x + c\right ) - 2048 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{12155 \, 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 )^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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