Optimal. Leaf size=22 \[ \frac {(a+b \sin (c+d x))^9}{9 b d} \]
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Rubi [A] time = 0.03, 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 = {2668, 32} \[ \frac {(a+b \sin (c+d x))^9}{9 b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^8 \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(a+b \sin (c+d x))^9}{9 b d}\\ \end {align*}
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Mathematica [B] time = 0.36, size = 137, normalized size = 6.23 \[ \frac {\sin (c+d x) \left (9 a^8+36 a^7 b \sin (c+d x)+84 a^6 b^2 \sin ^2(c+d x)+126 a^5 b^3 \sin ^3(c+d x)+126 a^4 b^4 \sin ^4(c+d x)+84 a^3 b^5 \sin ^5(c+d x)+36 a^2 b^6 \sin ^6(c+d x)+9 a b^7 \sin ^7(c+d x)+b^8 \sin ^8(c+d x)\right )}{9 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 257, normalized size = 11.68 \[ \frac {9 \, a b^{7} \cos \left (d x + c\right )^{8} - 12 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} + 18 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 36 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (b^{8} \cos \left (d x + c\right )^{8} + 9 \, a^{8} + 84 \, a^{6} b^{2} + 126 \, a^{4} b^{4} + 36 \, a^{2} b^{6} + b^{8} - 4 \, {\left (9 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (21 \, a^{4} b^{4} + 18 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (21 \, a^{6} b^{2} + 63 \, a^{4} b^{4} + 27 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.93, size = 20, normalized size = 0.91 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 21, normalized size = 0.95 \[ \frac {\left (a +b \sin \left (d x +c \right )\right )^{9}}{9 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 20, normalized size = 0.91 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 135, normalized size = 6.14 \[ \frac {a^8\,\sin \left (c+d\,x\right )+4\,a^7\,b\,{\sin \left (c+d\,x\right )}^2+\frac {28\,a^6\,b^2\,{\sin \left (c+d\,x\right )}^3}{3}+14\,a^5\,b^3\,{\sin \left (c+d\,x\right )}^4+14\,a^4\,b^4\,{\sin \left (c+d\,x\right )}^5+\frac {28\,a^3\,b^5\,{\sin \left (c+d\,x\right )}^6}{3}+4\,a^2\,b^6\,{\sin \left (c+d\,x\right )}^7+a\,b^7\,{\sin \left (c+d\,x\right )}^8+\frac {b^8\,{\sin \left (c+d\,x\right )}^9}{9}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.96, size = 168, normalized size = 7.64 \[ \begin {cases} \frac {a^{8} \sin {\left (c + d x \right )}}{d} + \frac {4 a^{7} b \sin ^{2}{\left (c + d x \right )}}{d} + \frac {28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {14 a^{5} b^{3} \sin ^{4}{\left (c + d x \right )}}{d} + \frac {14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )}}{d} + \frac {28 a^{3} b^{5} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )}}{d} + \frac {a b^{7} \sin ^{8}{\left (c + d x \right )}}{d} + \frac {b^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{8} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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