Optimal. Leaf size=63 \[ -\frac {8 a^2 \cos ^9(c+d x)}{99 d (a+a \sin (c+d x))^{9/2}}-\frac {2 a \cos ^9(c+d x)}{11 d (a+a \sin (c+d x))^{7/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752}
\begin {gather*} -\frac {8 a^2 \cos ^9(c+d x)}{99 d (a \sin (c+d x)+a)^{9/2}}-\frac {2 a \cos ^9(c+d x)}{11 d (a \sin (c+d x)+a)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 a \cos ^9(c+d x)}{11 d (a+a \sin (c+d x))^{7/2}}+\frac {1}{11} (4 a) \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^{7/2}} \, dx\\ &=-\frac {8 a^2 \cos ^9(c+d x)}{99 d (a+a \sin (c+d x))^{9/2}}-\frac {2 a \cos ^9(c+d x)}{11 d (a+a \sin (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 49, normalized size = 0.78 \begin {gather*} -\frac {2 \cos ^9(c+d x) (13+9 \sin (c+d x))}{99 d (1+\sin (c+d x))^2 (a (1+\sin (c+d x)))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 57, normalized size = 0.90
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right )^{5} \left (9 \sin \left (d x +c \right )+13\right )}{99 a^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs.
\(2 (55) = 110\).
time = 0.36, size = 161, normalized size = 2.56 \begin {gather*} -\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{6} - 23 \, \cos \left (d x + c\right )^{5} - 52 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} + {\left (9 \, \cos \left (d x + c\right )^{5} + 32 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} - 24 \, \cos \left (d x + c\right )^{2} - 32 \, \cos \left (d x + c\right ) - 64\right )} \sin \left (d x + c\right ) + 32 \, \cos \left (d x + c\right ) + 64\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{99 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \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 = 4.45, size = 68, normalized size = 1.08 \begin {gather*} -\frac {64 \, {\left (9 \, \sqrt {2} \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 11 \, \sqrt {2} \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}\right )}}{99 \, 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 = -0.02 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^8}{{\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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