Optimal. Leaf size=15 \[ \frac {\sin ^6(a+b x)}{6 b} \]
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Rubi [A]
time = 0.01, 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 = {2644, 30}
\begin {gather*} \frac {\sin ^6(a+b x)}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rubi steps
\begin {align*} \int \cos (a+b x) \sin ^5(a+b x) \, dx &=\frac {\text {Subst}\left (\int x^5 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {\sin ^6(a+b x)}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\sin ^6(a+b x)}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 14, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {\sin ^{6}\left (b x +a \right )}{6 b}\) | \(14\) |
default | \(\frac {\sin ^{6}\left (b x +a \right )}{6 b}\) | \(14\) |
norman | \(\frac {32 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{6}}\) | \(32\) |
risch | \(-\frac {\cos \left (6 b x +6 a \right )}{192 b}+\frac {\cos \left (4 b x +4 a \right )}{32 b}-\frac {5 \cos \left (2 b x +2 a \right )}{64 b}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} \frac {\sin \left (b x + a\right )^{6}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (13) = 26\).
time = 0.39, size = 34, normalized size = 2.27 \begin {gather*} -\frac {\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 20, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {\sin ^{6}{\left (a + b x \right )}}{6 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos {\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 = 3.48, size = 13, normalized size = 0.87 \begin {gather*} \frac {\sin \left (b x + a\right )^{6}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\sin \left (a+b\,x\right )}^6}{6\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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