Optimal. Leaf size=45 \[ \frac {(a+a \sin (c+d x))^{10}}{5 a^2 d}-\frac {(a+a \sin (c+d x))^{11}}{11 a^3 d} \]
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Rubi [A]
time = 0.03, 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 {(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^{11}}{11 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {\text {Subst}\left (\int (a-x) (a+x)^9 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {(a+a \sin (c+d x))^{10}}{5 a^2 d}-\frac {(a+a \sin (c+d x))^{11}}{11 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 43, normalized size = 0.96 \begin {gather*} -\frac {a^8 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{20} (-6+5 \sin (c+d x))}{55 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(462\) vs.
\(2(41)=82\).
time = 0.69, size = 463, normalized size = 10.29 Too large to display
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. 134 vs.
\(2 (41) = 82\).
time = 0.32, size = 134, normalized size = 2.98 \begin {gather*} -\frac {5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, 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. 136 vs.
\(2 (41) = 82\).
time = 0.39, size = 136, normalized size = 3.02 \begin {gather*} \frac {44 \, a^{8} \cos \left (d x + c\right )^{10} - 550 \, a^{8} \cos \left (d x + c\right )^{8} + 1760 \, a^{8} \cos \left (d x + c\right )^{6} - 1760 \, a^{8} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{8} \cos \left (d x + c\right )^{10} - 190 \, a^{8} \cos \left (d x + c\right )^{8} + 1040 \, a^{8} \cos \left (d x + c\right )^{6} - 1568 \, a^{8} \cos \left (d x + c\right )^{4} + 256 \, a^{8} \cos \left (d x + c\right )^{2} + 512 \, a^{8}\right )} \sin \left (d x + c\right )}{55 \, d} \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. 422 vs.
\(2 (36) = 72\).
time = 2.84, size = 422, normalized size = 9.38 \begin {gather*} \begin {cases} \frac {2 a^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac {8 a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} - \frac {2 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {56 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} - \frac {2 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {14 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {28 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {28 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {14 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a^{8} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac {7 a^{8} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac {14 a^{8} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {2 a^{8} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{8} \cos ^{3}{\left (c \right )} & \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. 134 vs.
\(2 (41) = 82\).
time = 6.63, size = 134, normalized size = 2.98 \begin {gather*} -\frac {5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 132, normalized size = 2.93 \begin {gather*} \frac {-\frac {a^8\,{\sin \left (c+d\,x\right )}^{11}}{11}-\frac {4\,a^8\,{\sin \left (c+d\,x\right )}^{10}}{5}-3\,a^8\,{\sin \left (c+d\,x\right )}^9-6\,a^8\,{\sin \left (c+d\,x\right )}^8-6\,a^8\,{\sin \left (c+d\,x\right )}^7+\frac {42\,a^8\,{\sin \left (c+d\,x\right )}^5}{5}+12\,a^8\,{\sin \left (c+d\,x\right )}^4+9\,a^8\,{\sin \left (c+d\,x\right )}^3+4\,a^8\,{\sin \left (c+d\,x\right )}^2+a^8\,\sin \left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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