Optimal. Leaf size=65 \[ -\frac {4}{5 a^3 d (a+a \sin (c+d x))^5}-\frac {1}{3 a^5 d (a+a \sin (c+d x))^3}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )^4} \]
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Rubi [A]
time = 0.04, 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 = {2746, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {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 = 0.08, size = 58, normalized size = 0.89 \begin {gather*} \frac {\cos ^6(c+d x) \left (2-5 \sin (c+d x)+5 \sin ^2(c+d x)\right )}{15 a^8 d (-1+\sin (c+d x))^3 (1+\sin (c+d x))^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 43, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {-\frac {4}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {1}{\left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}}{d \,a^{8}}\) | \(43\) |
default | \(\frac {-\frac {4}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {1}{\left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}}{d \,a^{8}}\) | \(43\) |
risch | \(\frac {8 i \left (-10 i {\mathrm e}^{6 i \left (d x +c \right )}+5 \,{\mathrm e}^{7 i \left (d x +c \right )}+10 i {\mathrm e}^{4 i \left (d x +c \right )}-18 \,{\mathrm e}^{5 i \left (d x +c \right )}+5 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{15 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10}}\) | \(82\) |
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 100, normalized size = 1.54 \begin {gather*} \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 )}} \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. 1120 vs.
\(2 (58) = 116\).
time = 14.30, size = 1120, normalized size = 17.23 \begin {gather*} \begin {cases} - \frac {8 \sin ^{4}{\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 {21 \sin ^{3}{\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 {12 \sin ^{2}{\left (c + d x \right )} \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 {19 \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 {14 \sin {\left (c + d x \right )} \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 {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 ^{4}{\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 {2 \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 ^{5}{\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 137 vs.
\(2 (61) = 122\).
time = 6.57, size = 137, normalized size = 2.11 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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