Optimal. Leaf size=121 \[ \frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^9 d}-\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^8 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{3 a^7 d}-\frac{64 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}+\frac{32 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0900153, antiderivative size = 121, 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} \[ \frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^9 d}-\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^8 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{3 a^7 d}-\frac{64 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}+\frac{32 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^9(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^4 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^9 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (16 a^4 (a+x)^{3/2}-32 a^3 (a+x)^{5/2}+24 a^2 (a+x)^{7/2}-8 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^9 d}\\ &=\frac{32 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}-\frac{64 (a+a \sin (c+d x))^{7/2}}{7 a^6 d}+\frac{16 (a+a \sin (c+d x))^{9/2}}{3 a^7 d}-\frac{16 (a+a \sin (c+d x))^{11/2}}{11 a^8 d}+\frac{2 (a+a \sin (c+d x))^{13/2}}{13 a^9 d}\\ \end{align*}
Mathematica [A] time = 0.290182, size = 64, normalized size = 0.53 \[ \frac{2 \left (1155 \sin ^4(c+d x)-6300 \sin ^3(c+d x)+14210 \sin ^2(c+d x)-16700 \sin (c+d x)+9683\right ) (a (\sin (c+d x)+1))^{5/2}}{15015 a^5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.102, size = 67, normalized size = 0.6 \begin{align*}{\frac{2310\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+12600\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -33040\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-46000\,\sin \left ( dx+c \right ) +50096}{15015\,{a}^{5}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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.942051, size = 120, normalized size = 0.99 \begin{align*} \frac{2 \,{\left (1155 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 10920 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 40040 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2} - 68640 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3} + 48048 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{4}\right )}}{15015 \, a^{9} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.30215, size = 247, normalized size = 2.04 \begin{align*} -\frac{2 \,{\left (1155 \, \cos \left (d x + c\right )^{6} - 6230 \, \cos \left (d x + c\right )^{4} - 512 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (1995 \, \cos \left (d x + c\right )^{4} - 1280 \, \cos \left (d x + c\right )^{2} - 2048\right )} \sin \left (d x + c\right ) - 4096\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15015 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31304, size = 120, normalized size = 0.99 \begin{align*} \frac{2 \,{\left (1155 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 10920 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 40040 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2} - 68640 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3} + 48048 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{4}\right )}}{15015 \, a^{9} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]