Optimal. Leaf size=27 \[ -\frac {\cos (a+b x)}{b}+\frac {\cos ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, 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 {\cos ^3(a+b x)}{3 b}-\frac {\cos (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \, dx &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos (a+b x)}{b}+\frac {\cos ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.07 \begin {gather*} -\frac {3 \cos (a+b x)}{4 b}+\frac {\cos (3 (a+b x))}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 22, normalized size = 0.81
method | result | size |
derivativedivides | \(-\frac {\left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}\) | \(22\) |
default | \(-\frac {\left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}\) | \(22\) |
risch | \(-\frac {3 \cos \left (b x +a \right )}{4 b}+\frac {\cos \left (3 b x +3 a \right )}{12 b}\) | \(27\) |
norman | \(\frac {-\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {4}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.36, size = 22, normalized size = 0.81 \begin {gather*} \frac {\cos \left (b x + a\right )^{3} - 3 \, \cos \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 = 0.79, size = 22, normalized size = 0.81 \begin {gather*} \frac {\cos \left (b x + a\right )^{3} - 3 \, \cos \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 = 37, normalized size = 1.37 \begin {gather*} \begin {cases} - \frac {\sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 \cos ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{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 = 0.72, size = 25, normalized size = 0.93 \begin {gather*} \frac {\cos \left (b x + a\right )^{3}}{3 \, b} - \frac {\cos \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 24, normalized size = 0.89 \begin {gather*} -\frac {3\,\cos \left (a+b\,x\right )-{\cos \left (a+b\,x\right )}^3}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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