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 = 0.09, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 43
Rule 2667
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 = 0.29, 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]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 82, normalized size = 0.68 \[ -\frac {2 \, {\left (1155 \, \cos \left (d x + c\right )^{6} - 6230 \, \cos \left (d x + c\right )^{4} - 512 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (1995 \, \cos \left (d x + c\right )^{4} - 1280 \, \cos \left (d x + c\right )^{2} - 2048\right )} \sin \left (d x + c\right ) - 4096\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{15015 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.95, size = 430, normalized size = 3.55 \[ \frac {2 \, {\left (\frac {9683 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {15015 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {25402 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {90090 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {107393 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {93093 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {183612 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {183612 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {93093 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {107393 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {90090 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {25402 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {9683 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {15015 \, a^{4}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15015 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {13}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 67, normalized size = 0.55 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \left (1155 \left (\cos ^{4}\left (d x +c \right )\right )+6300 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-16520 \left (\cos ^{2}\left (d x +c \right )\right )-23000 \sin \left (d x +c \right )+25048\right )}{15015 a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.67, size = 89, normalized size = 0.74 \[ \frac {2 \, {\left (1155 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 10920 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 40040 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2} - 68640 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{3} + 48048 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{4}\right )}}{15015 \, a^{9} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^9}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________