Optimal. Leaf size=59 \[ \frac{1}{2} \cos (2 a) \text{CosIntegral}(2 b x)+\frac{1}{8} \cos (4 a) \text{CosIntegral}(4 b x)-\frac{1}{2} \sin (2 a) \text{Si}(2 b x)-\frac{1}{8} \sin (4 a) \text{Si}(4 b x)+\frac{3 \log (x)}{8} \]
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Rubi [A] time = 0.157571, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3312, 3303, 3299, 3302} \[ \frac{1}{2} \cos (2 a) \text{CosIntegral}(2 b x)+\frac{1}{8} \cos (4 a) \text{CosIntegral}(4 b x)-\frac{1}{2} \sin (2 a) \text{Si}(2 b x)-\frac{1}{8} \sin (4 a) \text{Si}(4 b x)+\frac{3 \log (x)}{8} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\cos ^4(a+b x)}{x} \, dx &=\int \left (\frac{3}{8 x}+\frac{\cos (2 a+2 b x)}{2 x}+\frac{\cos (4 a+4 b x)}{8 x}\right ) \, dx\\ &=\frac{3 \log (x)}{8}+\frac{1}{8} \int \frac{\cos (4 a+4 b x)}{x} \, dx+\frac{1}{2} \int \frac{\cos (2 a+2 b x)}{x} \, dx\\ &=\frac{3 \log (x)}{8}+\frac{1}{2} \cos (2 a) \int \frac{\cos (2 b x)}{x} \, dx+\frac{1}{8} \cos (4 a) \int \frac{\cos (4 b x)}{x} \, dx-\frac{1}{2} \sin (2 a) \int \frac{\sin (2 b x)}{x} \, dx-\frac{1}{8} \sin (4 a) \int \frac{\sin (4 b x)}{x} \, dx\\ &=\frac{1}{2} \cos (2 a) \text{Ci}(2 b x)+\frac{1}{8} \cos (4 a) \text{Ci}(4 b x)+\frac{3 \log (x)}{8}-\frac{1}{2} \sin (2 a) \text{Si}(2 b x)-\frac{1}{8} \sin (4 a) \text{Si}(4 b x)\\ \end{align*}
Mathematica [A] time = 0.102006, size = 52, normalized size = 0.88 \[ \frac{1}{8} (4 \cos (2 a) \text{CosIntegral}(2 b x)+\cos (4 a) \text{CosIntegral}(4 b x)-4 \sin (2 a) \text{Si}(2 b x)-\sin (4 a) \text{Si}(4 b x)+3 \log (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 52, normalized size = 0.9 \begin{align*} -{\frac{{\it Si} \left ( 4\,bx \right ) \sin \left ( 4\,a \right ) }{8}}+{\frac{{\it Ci} \left ( 4\,bx \right ) \cos \left ( 4\,a \right ) }{8}}-{\frac{{\it Si} \left ( 2\,bx \right ) \sin \left ( 2\,a \right ) }{2}}+{\frac{{\it Ci} \left ( 2\,bx \right ) \cos \left ( 2\,a \right ) }{2}}+{\frac{3\,\ln \left ( bx \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.20682, size = 123, normalized size = 2.08 \begin{align*} -\frac{1}{16} \,{\left (E_{1}\left (4 i \, b x\right ) + E_{1}\left (-4 i \, b x\right )\right )} \cos \left (4 \, a\right ) - \frac{1}{4} \,{\left (E_{1}\left (2 i \, b x\right ) + E_{1}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + \frac{1}{16} \,{\left (i \, E_{1}\left (4 i \, b x\right ) - i \, E_{1}\left (-4 i \, b x\right )\right )} \sin \left (4 \, a\right ) + \frac{1}{16} \,{\left (4 i \, E_{1}\left (2 i \, b x\right ) - 4 i \, E_{1}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right ) + \frac{3}{8} \, \log \left (b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47171, size = 274, normalized size = 4.64 \begin{align*} \frac{1}{16} \,{\left (\operatorname{Ci}\left (4 \, b x\right ) + \operatorname{Ci}\left (-4 \, b x\right )\right )} \cos \left (4 \, a\right ) + \frac{1}{4} \,{\left (\operatorname{Ci}\left (2 \, b x\right ) + \operatorname{Ci}\left (-2 \, b x\right )\right )} \cos \left (2 \, a\right ) - \frac{1}{8} \, \sin \left (4 \, a\right ) \operatorname{Si}\left (4 \, b x\right ) - \frac{1}{2} \, \sin \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x\right ) + \frac{3}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.73744, size = 60, normalized size = 1.02 \begin{align*} \frac{3 \log{\left (x \right )}}{8} - \frac{\sin{\left (2 a \right )} \operatorname{Si}{\left (2 b x \right )}}{2} - \frac{\sin{\left (4 a \right )} \operatorname{Si}{\left (4 b x \right )}}{8} + \frac{\cos{\left (2 a \right )} \operatorname{Ci}{\left (2 b x \right )}}{2} + \frac{\cos{\left (4 a \right )} \operatorname{Ci}{\left (4 b x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.14128, size = 578, normalized size = 9.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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