Optimal. Leaf size=45 \[ \frac {1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {1}{3 a^2 d (a \sin (c+d x)+a)^6} \]
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Rubi [A] time = 0.05, antiderivative size = 45, 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} \[ \frac {1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac {1}{3 a^2 d (a \sin (c+d x)+a)^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{(a+x)^7} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {2 a}{(a+x)^7}-\frac {1}{(a+x)^6}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {1}{3 a^2 d (a+a \sin (c+d x))^6}+\frac {1}{5 a^3 d (a+a \sin (c+d x))^5}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 43, normalized size = 0.96 \[ \frac {3 \sin (c+d x)-2}{15 a^8 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{12}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 105, normalized size = 2.33 \[ -\frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, {\left (a^{8} d \cos \left (d x + c\right )^{6} - 18 \, a^{8} d \cos \left (d x + c\right )^{4} + 48 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d - 2 \, {\left (3 \, a^{8} d \cos \left (d x + c\right )^{4} - 16 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 28, normalized size = 0.62 \[ \frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, a^{8} d {\left (\sin \left (d x + c\right ) + 1\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 33, normalized size = 0.73 \[ \frac {\frac {1}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{6}}}{d \,a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 96, normalized size = 2.13 \[ \frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, {\left (a^{8} \sin \left (d x + c\right )^{6} + 6 \, a^{8} \sin \left (d x + c\right )^{5} + 15 \, a^{8} \sin \left (d x + c\right )^{4} + 20 \, a^{8} \sin \left (d x + c\right )^{3} + 15 \, a^{8} \sin \left (d x + c\right )^{2} + 6 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 28, normalized size = 0.62 \[ \frac {3\,\sin \left (c+d\,x\right )-2}{15\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.54, size = 493, normalized size = 10.96 \[ \begin {cases} \frac {6 \sin ^{2}{\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} + \frac {7 \sin {\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} - \frac {15 \cos ^{2}{\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} + \frac {1}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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