Optimal. Leaf size=97 \[ -\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d} \]
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Rubi [A] time = 0.0829884, 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} \[ -\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^7(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{7/2}-12 a^2 (a+x)^{9/2}+6 a (a+x)^{11/2}-(a+x)^{13/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{16 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}-\frac{24 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}+\frac{12 (a+a \sin (c+d x))^{13/2}}{13 a^6 d}-\frac{2 (a+a \sin (c+d x))^{15/2}}{15 a^7 d}\\ \end{align*}
Mathematica [A] time = 4.2383, size = 74, normalized size = 0.76 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^8 (-10755 \sin (c+d x)+429 \sin (3 (c+d x))-3366 \cos (2 (c+d x))+8330)}{12870 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 57, normalized size = 0.6 \begin{align*}{\frac{858\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -3366\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-5592\,\sin \left ( dx+c \right ) +5848}{6435\,{a}^{4}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 = 0.948102, size = 97, normalized size = 1. \begin{align*} -\frac{2 \,{\left (429 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{15}{2}} - 2970 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a + 7020 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a^{2} - 5720 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{3}\right )}}{6435 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74841, size = 254, normalized size = 2.62 \begin{align*} \frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{6} + 56 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} +{\left (429 \, \cos \left (d x + c\right )^{6} + 504 \, \cos \left (d x + c\right )^{4} + 640 \, \cos \left (d x + c\right )^{2} + 1024\right )} \sin \left (d x + c\right ) + 1024\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{6435 \, 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 [A] time = 2.16911, size = 104, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (\frac{429 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{15}{2}}}{a^{6}} - \frac{2970 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}}}{a^{5}} + \frac{7020 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}}}{a^{4}} - \frac{5720 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}}}{a^{3}}\right )}}{6435 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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