Optimal. Leaf size=15 \[ \frac {\sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2564, 30} \[ \frac {\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rubi steps
\begin {align*} \int \cos (a+b x) \sin ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 21, normalized size = 1.40 \[ -\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 13, normalized size = 0.87 \[ \frac {\sin \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.93 \[ \frac {\sin ^{3}\left (b x +a \right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 13, normalized size = 0.87 \[ \frac {\sin \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 13, normalized size = 0.87 \[ \frac {{\sin \left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 20, normalized size = 1.33 \[ \begin {cases} \frac {\sin ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\relax (a )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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