Optimal. Leaf size=37 \[ \frac{1}{4} \cos (2 a) \text{CosIntegral}\left (2 b x^2\right )-\frac{1}{4} \sin (2 a) \text{Si}\left (2 b x^2\right )+\frac{\log (x)}{2} \]
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Rubi [A] time = 0.0518501, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3404, 3378, 3376, 3375} \[ \frac{1}{4} \cos (2 a) \text{CosIntegral}\left (2 b x^2\right )-\frac{1}{4} \sin (2 a) \text{Si}\left (2 b x^2\right )+\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 3404
Rule 3378
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\cos ^2\left (a+b x^2\right )}{x} \, dx &=\int \left (\frac{1}{2 x}+\frac{\cos \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \int \frac{\cos \left (2 a+2 b x^2\right )}{x} \, dx\\ &=\frac{\log (x)}{2}+\frac{1}{2} \cos (2 a) \int \frac{\cos \left (2 b x^2\right )}{x} \, dx-\frac{1}{2} \sin (2 a) \int \frac{\sin \left (2 b x^2\right )}{x} \, dx\\ &=\frac{1}{4} \cos (2 a) \text{Ci}\left (2 b x^2\right )+\frac{\log (x)}{2}-\frac{1}{4} \sin (2 a) \text{Si}\left (2 b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0672536, size = 34, normalized size = 0.92 \[ \frac{1}{4} \left (\cos (2 a) \text{CosIntegral}\left (2 b x^2\right )-\sin (2 a) \text{Si}\left (2 b x^2\right )+2 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 32, normalized size = 0.9 \begin{align*}{\frac{{\it Ci} \left ( 2\,b{x}^{2} \right ) \cos \left ( 2\,a \right ) }{4}}+{\frac{\ln \left ( x \right ) }{2}}-{\frac{{\it Si} \left ( 2\,b{x}^{2} \right ) \sin \left ( 2\,a \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.36611, size = 69, normalized size = 1.86 \begin{align*} \frac{1}{8} \,{\left ({\rm Ei}\left (2 i \, b x^{2}\right ) +{\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + \frac{1}{8} \,{\left (i \,{\rm Ei}\left (2 i \, b x^{2}\right ) - i \,{\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60196, size = 153, normalized size = 4.14 \begin{align*} \frac{1}{8} \,{\left (\operatorname{Ci}\left (2 \, b x^{2}\right ) + \operatorname{Ci}\left (-2 \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) - \frac{1}{4} \, \sin \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x^{2}\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (a + b x^{2} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16839, size = 47, normalized size = 1.27 \begin{align*} \frac{1}{4} \, \cos \left (2 \, a\right ) \operatorname{Ci}\left (2 \, b x^{2}\right ) + \frac{1}{4} \, \sin \left (2 \, a\right ) \operatorname{Si}\left (-2 \, b x^{2}\right ) + \frac{1}{4} \, \log \left (b x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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