Optimal. Leaf size=97 \[ -\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d} \]
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Rubi [A] time = 0.0822959, 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} \[ -\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{3/2}-12 a^2 (a+x)^{5/2}+6 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{16 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}-\frac{24 (a+a \sin (c+d x))^{7/2}}{7 a^5 d}+\frac{4 (a+a \sin (c+d x))^{9/2}}{3 a^6 d}-\frac{2 (a+a \sin (c+d x))^{11/2}}{11 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.225519, size = 54, normalized size = 0.56 \[ -\frac{2 \left (105 \sin ^3(c+d x)-455 \sin ^2(c+d x)+755 \sin (c+d x)-533\right ) (a (\sin (c+d x)+1))^{5/2}}{1155 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 57, normalized size = 0.6 \begin{align*}{\frac{210\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -910\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1720\,\sin \left ( dx+c \right ) +1976}{1155\,{a}^{4}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973599, size = 97, normalized size = 1. \begin{align*} -\frac{2 \,{\left (105 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} - 770 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a + 1980 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{2} - 1848 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{3}\right )}}{1155 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29302, size = 204, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (245 \, \cos \left (d x + c\right )^{4} + 32 \, \cos \left (d x + c\right )^{2} -{\left (105 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{2} - 256\right )} \sin \left (d x + c\right ) + 256\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{1155 \, a^{2} 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 = 1.25644, size = 97, normalized size = 1. \begin{align*} -\frac{2 \,{\left (105 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} - 770 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a + 1980 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{2} - 1848 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{3}\right )}}{1155 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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