Optimal. Leaf size=121 \[ \frac {32 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}-\frac {64 (a+a \sin (c+d x))^{7/2}}{7 a^6 d}+\frac {16 (a+a \sin (c+d x))^{9/2}}{3 a^7 d}-\frac {16 (a+a \sin (c+d x))^{11/2}}{11 a^8 d}+\frac {2 (a+a \sin (c+d x))^{13/2}}{13 a^9 d} \]
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Rubi [A]
time = 0.06, 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 = {2746, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^9(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\text {Subst}\left (\int (a-x)^4 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^9 d}\\ &=\frac {\text {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*}
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Mathematica [A]
time = 0.18, size = 64, normalized size = 0.53 \begin {gather*} \frac {2 (a (1+\sin (c+d x)))^{5/2} \left (9683-16700 \sin (c+d x)+14210 \sin ^2(c+d x)-6300 \sin ^3(c+d x)+1155 \sin ^4(c+d x)\right )}{15015 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 67, normalized size = 0.55
method | result | size |
default | \(\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \left (1155 \left (\cos ^{4}\left (d x +c \right )\right )+6300 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-16520 \left (\cos ^{2}\left (d x +c \right )\right )-23000 \sin \left (d x +c \right )+25048\right )}{15015 a^{5} d}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 89, normalized size = 0.74 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 82, normalized size = 0.68 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.48, size = 134, normalized size = 1.11 \begin {gather*} \frac {128 \, {\left (1155 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 5460 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 10010 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 8580 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3003 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}\right )}}{15015 \, a^{3} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^9}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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