Optimal. Leaf size=25 \[ \frac{1}{2} \cos (a) \text{CosIntegral}\left (b x^2\right )-\frac{1}{2} \sin (a) \text{Si}\left (b x^2\right ) \]
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Rubi [A] time = 0.0289689, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3378, 3376, 3375} \[ \frac{1}{2} \cos (a) \text{CosIntegral}\left (b x^2\right )-\frac{1}{2} \sin (a) \text{Si}\left (b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 3378
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b x^2\right )}{x} \, dx &=\cos (a) \int \frac{\cos \left (b x^2\right )}{x} \, dx-\sin (a) \int \frac{\sin \left (b x^2\right )}{x} \, dx\\ &=\frac{1}{2} \cos (a) \text{Ci}\left (b x^2\right )-\frac{1}{2} \sin (a) \text{Si}\left (b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0544486, size = 24, normalized size = 0.96 \[ \frac{1}{2} \left (\cos (a) \text{CosIntegral}\left (b x^2\right )-\sin (a) \text{Si}\left (b x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 22, normalized size = 0.9 \begin{align*}{\frac{{\it Ci} \left ( b{x}^{2} \right ) \cos \left ( a \right ) }{2}}-{\frac{{\it Si} \left ( b{x}^{2} \right ) \sin \left ( a \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.46817, size = 58, normalized size = 2.32 \begin{align*} \frac{1}{4} \,{\left ({\rm Ei}\left (i \, b x^{2}\right ) +{\rm Ei}\left (-i \, b x^{2}\right )\right )} \cos \left (a\right ) + \frac{1}{4} \,{\left (i \,{\rm Ei}\left (i \, b x^{2}\right ) - i \,{\rm Ei}\left (-i \, b x^{2}\right )\right )} \sin \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53231, size = 122, normalized size = 4.88 \begin{align*} \frac{1}{4} \,{\left (\operatorname{Ci}\left (b x^{2}\right ) + \operatorname{Ci}\left (-b x^{2}\right )\right )} \cos \left (a\right ) - \frac{1}{2} \, \sin \left (a\right ) \operatorname{Si}\left (b x^{2}\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{\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 = 1.1287, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{2} \, \cos \left (a\right ) \operatorname{Ci}\left (b x^{2}\right ) - \frac{1}{2} \, \sin \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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