Optimal. Leaf size=65 \[ -\frac {1}{3 a^5 d (a \sin (c+d x)+a)^3}-\frac {4}{5 a^3 d (a \sin (c+d x)+a)^5}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )^4} \]
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Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ -\frac {1}{3 a^5 d (a \sin (c+d x)+a)^3}+\frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {4}{5 a^3 d (a \sin (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{(a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^6}-\frac {4 a}{(a+x)^5}+\frac {1}{(a+x)^4}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {4}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {1}{3 a^5 d (a+a \sin (c+d x))^3}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )^4}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 58, normalized size = 0.89 \[ \frac {\left (5 \sin ^2(c+d x)-5 \sin (c+d x)+2\right ) \cos ^6(c+d x)}{15 a^8 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 100, normalized size = 1.54 \[ \frac {5 \, \cos \left (d x + c\right )^{2} + 5 \, \sin \left (d x + c\right ) - 7}{15 \, {\left (5 \, a^{8} d \cos \left (d x + c\right )^{4} - 20 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d + {\left (a^{8} d \cos \left (d x + c\right )^{4} - 12 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.44, size = 137, normalized size = 2.11 \[ \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 282 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 43, normalized size = 0.66 \[ \frac {-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{\left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {4}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}}{d \,a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 93, normalized size = 1.43 \[ -\frac {5 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{15 \, {\left (a^{8} \sin \left (d x + c\right )^{5} + 5 \, a^{8} \sin \left (d x + c\right )^{4} + 10 \, a^{8} \sin \left (d x + c\right )^{3} + 10 \, a^{8} \sin \left (d x + c\right )^{2} + 5 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.71, size = 54, normalized size = 0.83 \[ \frac {1}{a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^4}-\frac {1}{3\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3}-\frac {4}{5\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 42.11, size = 1120, normalized size = 17.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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