Optimal. Leaf size=22 \[ \frac {(a+b \sin (c+d x))^3}{3 b d} \]
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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} \[ \frac {(a+b \sin (c+d x))^3}{3 b d} \]
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))^2 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^2 \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(a+b \sin (c+d x))^3}{3 b d}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 46, normalized size = 2.09 \[ \frac {a^2 \sin (c+d x)}{d}+\frac {a b \sin ^2(c+d x)}{d}+\frac {b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 48, normalized size = 2.18 \[ -\frac {3 \, a b \cos \left (d x + c\right )^{2} + {\left (b^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 20, normalized size = 0.91 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \]
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 \[ \frac {\left (a +b \sin \left (d x +c \right )\right )^{3}}{3 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 20, normalized size = 0.91 \[ \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 39, normalized size = 1.77 \[ \frac {a^2\,\sin \left (c+d\,x\right )+a\,b\,{\sin \left (c+d\,x\right )}^2+\frac {b^2\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 53, normalized size = 2.41 \[ \begin {cases} \frac {a^{2} \sin {\left (c + d x \right )}}{d} + \frac {a b \sin ^{2}{\left (c + d x \right )}}{d} + \frac {b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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