Optimal. Leaf size=67 \[ \frac {x}{16}+\frac {\cos (a+b x) \sin (a+b x)}{16 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{24 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2648, 2715, 8}
\begin {gather*} -\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {\sin (a+b x) \cos ^3(a+b x)}{24 b}+\frac {\sin (a+b x) \cos (a+b x)}{16 b}+\frac {x}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
Rubi steps
\begin {align*} \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx &=-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac {1}{6} \int \cos ^4(a+b x) \, dx\\ &=\frac {\cos ^3(a+b x) \sin (a+b x)}{24 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac {1}{8} \int \cos ^2(a+b x) \, dx\\ &=\frac {\cos (a+b x) \sin (a+b x)}{16 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{24 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b}+\frac {\int 1 \, dx}{16}\\ &=\frac {x}{16}+\frac {\cos (a+b x) \sin (a+b x)}{16 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{24 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.60 \begin {gather*} -\frac {-12 b x-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))}{192 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 54, normalized size = 0.81
method | result | size |
risch | \(\frac {x}{16}-\frac {\sin \left (6 b x +6 a \right )}{192 b}-\frac {\sin \left (4 b x +4 a \right )}{64 b}+\frac {\sin \left (2 b x +2 a \right )}{64 b}\) | \(47\) |
derivativedivides | \(\frac {-\frac {\left (\cos ^{5}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{6}+\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{24}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(54\) |
default | \(\frac {-\frac {\left (\cos ^{5}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{6}+\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{24}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(54\) |
norman | \(\frac {\frac {x}{16}-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {47 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}-\frac {13 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {13 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}-\frac {47 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {3 x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {15 x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {5 x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {15 x \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {3 x \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {x \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{6}}\) | \(199\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 37, normalized size = 0.55 \begin {gather*} \frac {4 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 12 \, b x + 12 \, a - 3 \, \sin \left (4 \, b x + 4 \, a\right )}{192 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 47, normalized size = 0.70 \begin {gather*} \frac {3 \, b x - {\left (8 \, \cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{48 \, 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. 136 vs.
\(2 (56) = 112\).
time = 0.40, size = 136, normalized size = 2.03 \begin {gather*} \begin {cases} \frac {x \sin ^{6}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {x \cos ^{6}{\left (a + b x \right )}}{16} + \frac {\sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{16 b} + \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{6 b} - \frac {\sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{4}{\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 = 6.04, size = 46, normalized size = 0.69 \begin {gather*} \frac {1}{16} \, x - \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{64 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 43, normalized size = 0.64 \begin {gather*} \frac {x}{16}-\frac {\frac {\sin \left (4\,a+4\,b\,x\right )}{64}-\frac {\sin \left (2\,a+2\,b\,x\right )}{64}+\frac {\sin \left (6\,a+6\,b\,x\right )}{192}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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