Optimal. Leaf size=22 \[ \frac {(a \sin (c+d x)+a)^9}{9 a d} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac {(a \sin (c+d x)+a)^9}{9 a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^8 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {(a+a \sin (c+d x))^9}{9 a d}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 147, normalized size = 6.68 \[ \frac {a^8 \sin ^9(c+d x)}{9 d}+\frac {a^8 \sin ^8(c+d x)}{d}+\frac {4 a^8 \sin ^7(c+d x)}{d}+\frac {28 a^8 \sin ^6(c+d x)}{3 d}+\frac {14 a^8 \sin ^5(c+d x)}{d}+\frac {14 a^8 \sin ^4(c+d x)}{d}+\frac {28 a^8 \sin ^3(c+d x)}{3 d}+\frac {4 a^8 \sin ^2(c+d x)}{d}+\frac {a^8 \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 122, normalized size = 5.55 \[ \frac {9 \, a^{8} \cos \left (d x + c\right )^{8} - 120 \, a^{8} \cos \left (d x + c\right )^{6} + 432 \, a^{8} \cos \left (d x + c\right )^{4} - 576 \, a^{8} \cos \left (d x + c\right )^{2} + {\left (a^{8} \cos \left (d x + c\right )^{8} - 40 \, a^{8} \cos \left (d x + c\right )^{6} + 240 \, a^{8} \cos \left (d x + c\right )^{4} - 448 \, a^{8} \cos \left (d x + c\right )^{2} + 256 \, a^{8}\right )} \sin \left (d x + c\right )}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 20, normalized size = 0.91 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 21, normalized size = 0.95 \[ \frac {\left (a +a \sin \left (d x +c \right )\right )^{9}}{9 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 20, normalized size = 0.91 \[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.75, size = 118, normalized size = 5.36 \[ \frac {\frac {a^8\,{\sin \left (c+d\,x\right )}^9}{9}+a^8\,{\sin \left (c+d\,x\right )}^8+4\,a^8\,{\sin \left (c+d\,x\right )}^7+\frac {28\,a^8\,{\sin \left (c+d\,x\right )}^6}{3}+14\,a^8\,{\sin \left (c+d\,x\right )}^5+14\,a^8\,{\sin \left (c+d\,x\right )}^4+\frac {28\,a^8\,{\sin \left (c+d\,x\right )}^3}{3}+4\,a^8\,{\sin \left (c+d\,x\right )}^2+a^8\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.84, size = 148, normalized size = 6.73 \[ \begin {cases} \frac {a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {a^{8} \sin ^{8}{\left (c + d x \right )}}{d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} + \frac {28 a^{8} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {14 a^{8} \sin ^{5}{\left (c + d x \right )}}{d} + \frac {14 a^{8} \sin ^{4}{\left (c + d x \right )}}{d} + \frac {28 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 a^{8} \sin ^{2}{\left (c + d x \right )}}{d} + \frac {a^{8} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{8} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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