Optimal. Leaf size=80 \[ \frac {3 x^2}{16}+\frac {3 \cos ^2(a+b x)}{16 b^2}+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {3 x \cos (a+b x) \sin (a+b x)}{8 b}+\frac {x \cos ^3(a+b x) \sin (a+b x)}{4 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3391, 30}
\begin {gather*} \frac {\cos ^4(a+b x)}{16 b^2}+\frac {3 \cos ^2(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3 x \sin (a+b x) \cos (a+b x)}{8 b}+\frac {3 x^2}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3391
Rubi steps
\begin {align*} \int x \cos ^4(a+b x) \, dx &=\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac {3}{4} \int x \cos ^2(a+b x) \, dx\\ &=\frac {3 \cos ^2(a+b x)}{16 b^2}+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {3 x \cos (a+b x) \sin (a+b x)}{8 b}+\frac {x \cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac {3 \int x \, dx}{8}\\ &=\frac {3 x^2}{16}+\frac {3 \cos ^2(a+b x)}{16 b^2}+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {3 x \cos (a+b x) \sin (a+b x)}{8 b}+\frac {x \cos ^3(a+b x) \sin (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 53, normalized size = 0.66 \begin {gather*} \frac {16 \cos (2 (a+b x))+\cos (4 (a+b x))+4 b x (6 b x+8 \sin (2 (a+b x))+\sin (4 (a+b x)))}{128 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 106, normalized size = 1.32
method | result | size |
risch | \(\frac {3 x^{2}}{16}+\frac {\cos \left (4 b x +4 a \right )}{128 b^{2}}+\frac {x \sin \left (4 b x +4 a \right )}{32 b}+\frac {\cos \left (2 b x +2 a \right )}{8 b^{2}}+\frac {x \sin \left (2 b x +2 a \right )}{4 b}\) | \(65\) |
derivativedivides | \(\frac {\left (b x +a \right ) \left (\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-\frac {3 \left (b x +a \right )^{2}}{16}+\frac {\left (2 \left (\cos ^{2}\left (b x +a \right )\right )+3\right )^{2}}{64}-a \left (\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{b^{2}}\) | \(106\) |
default | \(\frac {\left (b x +a \right ) \left (\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-\frac {3 \left (b x +a \right )^{2}}{16}+\frac {\left (2 \left (\cos ^{2}\left (b x +a \right )\right )+3\right )^{2}}{64}-a \left (\frac {\left (\cos ^{3}\left (b x +a \right )+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{b^{2}}\) | \(106\) |
norman | \(\frac {\frac {3 x^{2}}{16}+\frac {3 x^{2} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {9 x^{2} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {3 x^{2} \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {3 x^{2} \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{16}+\frac {5 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{4 b}-\frac {3 x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {3 x \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}-\frac {5 x \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}-\frac {3 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{2}}-\frac {5 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b^{2}}-\frac {5 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b^{2}}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4}}\) | \(201\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 98, normalized size = 1.22 \begin {gather*} \frac {24 \, {\left (b x + a\right )}^{2} - 4 \, {\left (12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a + 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right ) + 16 \, \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 63, normalized size = 0.79 \begin {gather*} \frac {3 \, b^{2} x^{2} + \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} + 2 \, {\left (2 \, b x \cos \left (b x + a\right )^{3} + 3 \, b x \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{16 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 138, normalized size = 1.72 \begin {gather*} \begin {cases} \frac {3 x^{2} \sin ^{4}{\left (a + b x \right )}}{16} + \frac {3 x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8} + \frac {3 x^{2} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} + \frac {5 x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {3 \sin ^{4}{\left (a + b x \right )}}{32 b^{2}} + \frac {5 \cos ^{4}{\left (a + b x \right )}}{32 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \cos ^{4}{\left (a \right )}}{2} & \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 = 64, normalized size = 0.80 \begin {gather*} \frac {3}{16} \, x^{2} + \frac {x \sin \left (4 \, b x + 4 \, a\right )}{32 \, b} + \frac {x \sin \left (2 \, b x + 2 \, a\right )}{4 \, b} + \frac {\cos \left (4 \, b x + 4 \, a\right )}{128 \, b^{2}} + \frac {\cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 63, normalized size = 0.79 \begin {gather*} \frac {3\,x^2}{16}-\frac {\frac {{\sin \left (2\,a+2\,b\,x\right )}^2}{64}-b\,\left (\frac {x\,\sin \left (2\,a+2\,b\,x\right )}{4}+\frac {x\,\sin \left (4\,a+4\,b\,x\right )}{32}\right )+\frac {{\sin \left (a+b\,x\right )}^2}{4}}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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