Optimal. Leaf size=26 \[ \frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713}
\begin {gather*} \frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {\sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 22, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3 b}\) | \(22\) |
default | \(\frac {\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3 b}\) | \(22\) |
risch | \(\frac {3 \sin \left (b x +a \right )}{4 b}+\frac {\sin \left (3 b x +3 a \right )}{12 b}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.53, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 21, normalized size = 0.81 \begin {gather*} \frac {{\left (\cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 36, normalized size = 1.38 \begin {gather*} \begin {cases} \frac {2 \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {\sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 0.92 \begin {gather*} \frac {3\,\sin \left (a+b\,x\right )-{\sin \left (a+b\,x\right )}^3}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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