Optimal. Leaf size=45 \[ -\frac {1}{3 a^2 d (a+a \sin (c+d x))^6}+\frac {1}{5 a^3 d (a+a \sin (c+d x))^5} \]
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Rubi [A]
time = 0.04, 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 = {2746, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\text {Subst}\left (\int \frac {a-x}{(a+x)^7} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\text {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 = 0.11, size = 43, normalized size = 0.96 \begin {gather*} \frac {-2+3 \sin (c+d x)}{15 a^8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 33, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{6}}+\frac {1}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}}{d \,a^{8}}\) | \(33\) |
default | \(\frac {-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{6}}+\frac {1}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}}{d \,a^{8}}\) | \(33\) |
risch | \(\frac {32 i \left (-4 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-3 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{15 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{12}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs.
\(2 (41) = 82\).
time = 0.28, size = 96, normalized size = 2.13 \begin {gather*} \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} \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. 105 vs.
\(2 (41) = 82\).
time = 0.35, size = 105, normalized size = 2.33 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 493 vs.
\(2 (39) = 78\).
time = 14.51, size = 493, normalized size = 10.96 \begin {gather*} \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}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.49, size = 28, normalized size = 0.62 \begin {gather*} \frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, a^{8} d {\left (\sin \left (d x + c\right ) + 1\right )}^{6}} \end {gather*}
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 \begin {gather*} \frac {3\,\sin \left (c+d\,x\right )-2}{15\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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