Optimal. Leaf size=15 \[ -\frac {\cos ^4(a+b x)}{2 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4372, 2645, 30}
\begin {gather*} -\frac {\cos ^4(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2645
Rule 4372
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cos ^3(a+b x) \sin (a+b x) \, dx\\ &=-\frac {2 \text {Subst}\left (\int x^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^4(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {\cos ^4(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs.
\(2(13)=26\).
time = 0.14, size = 30, normalized size = 2.00
method | result | size |
default | \(-\frac {\cos \left (2 x b +2 a \right )}{4 b}-\frac {\cos \left (4 x b +4 a \right )}{16 b}\) | \(30\) |
risch | \(-\frac {\cos \left (2 x b +2 a \right )}{4 b}-\frac {\cos \left (4 x b +4 a \right )}{16 b}\) | \(30\) |
norman | \(\frac {x \left (\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+x \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan ^{2}\left (x b +a \right )\right )+\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \tan \left (x b +a \right )}{b}-x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+\frac {x \tan \left (x b +a \right )}{2}-3 x \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right ) \tan \left (x b +a \right )-x \left (\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )\right ) \left (\tan ^{2}\left (x b +a \right )\right )+\frac {x \left (\tan ^{4}\left (\frac {a}{2}+\frac {x b}{2}\right )\right ) \tan \left (x b +a \right )}{2}-\frac {2 \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{b}+\frac {2 \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right ) \left (\tan ^{2}\left (x b +a \right )\right )}{b}-\frac {3 \left (\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )\right ) \tan \left (x b +a \right )}{b}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )^{2} \left (1+\tan ^{2}\left (x b +a \right )\right )}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 1.73 \begin {gather*} -\frac {\cos \left (4 \, b x + 4 \, a\right ) + 4 \, \cos \left (2 \, b x + 2 \, a\right )}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.71, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\cos \left (b x + a\right )^{4}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs.
\(2 (12) = 24\).
time = 0.43, size = 133, normalized size = 8.87 \begin {gather*} \begin {cases} - \frac {x \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )}}{4} - \frac {x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{2} + \frac {x \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} - \frac {\sin ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{2 b} + \frac {3 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x \sin {\left (2 a \right )} \cos ^{2}{\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 = 0.41, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\cos \left (b x + a\right )^{4}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 13, normalized size = 0.87 \begin {gather*} -\frac {{\cos \left (a+b\,x\right )}^4}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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