Optimal. Leaf size=55 \[ \frac{3}{8} \cos (a) \text{CosIntegral}\left (b x^2\right )+\frac{1}{8} \cos (3 a) \text{CosIntegral}\left (3 b x^2\right )-\frac{3}{8} \sin (a) \text{Si}\left (b x^2\right )-\frac{1}{8} \sin (3 a) \text{Si}\left (3 b x^2\right ) \]
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Rubi [A] time = 0.0832228, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3404, 3378, 3376, 3375} \[ \frac{3}{8} \cos (a) \text{CosIntegral}\left (b x^2\right )+\frac{1}{8} \cos (3 a) \text{CosIntegral}\left (3 b x^2\right )-\frac{3}{8} \sin (a) \text{Si}\left (b x^2\right )-\frac{1}{8} \sin (3 a) \text{Si}\left (3 b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 3404
Rule 3378
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\cos ^3\left (a+b x^2\right )}{x} \, dx &=\int \left (\frac{3 \cos \left (a+b x^2\right )}{4 x}+\frac{\cos \left (3 a+3 b x^2\right )}{4 x}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cos \left (3 a+3 b x^2\right )}{x} \, dx+\frac{3}{4} \int \frac{\cos \left (a+b x^2\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cos (a)) \int \frac{\cos \left (b x^2\right )}{x} \, dx+\frac{1}{4} \cos (3 a) \int \frac{\cos \left (3 b x^2\right )}{x} \, dx-\frac{1}{4} (3 \sin (a)) \int \frac{\sin \left (b x^2\right )}{x} \, dx-\frac{1}{4} \sin (3 a) \int \frac{\sin \left (3 b x^2\right )}{x} \, dx\\ &=\frac{3}{8} \cos (a) \text{Ci}\left (b x^2\right )+\frac{1}{8} \cos (3 a) \text{Ci}\left (3 b x^2\right )-\frac{3}{8} \sin (a) \text{Si}\left (b x^2\right )-\frac{1}{8} \sin (3 a) \text{Si}\left (3 b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.104799, size = 50, normalized size = 0.91 \[ \frac{1}{8} \left (3 \cos (a) \text{CosIntegral}\left (b x^2\right )+\cos (3 a) \text{CosIntegral}\left (3 b x^2\right )-3 \sin (a) \text{Si}\left (b x^2\right )-\sin (3 a) \text{Si}\left (3 b x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.103, size = 125, normalized size = 2.3 \begin{align*}{\frac{i}{16}}{{\rm e}^{-3\,ia}}\pi \,{\it csgn} \left ( b{x}^{2} \right ) -{\frac{i}{8}}{{\rm e}^{-3\,ia}}{\it Si} \left ( 3\,b{x}^{2} \right ) -{\frac{{{\rm e}^{-3\,ia}}{\it Ei} \left ( 1,-3\,ib{x}^{2} \right ) }{16}}+{\frac{3\,i}{16}}\pi \,{\it csgn} \left ( b{x}^{2} \right ){{\rm e}^{-ia}}-{\frac{3\,i}{8}}{{\rm e}^{-ia}}{\it Si} \left ( b{x}^{2} \right ) -{\frac{3\,{{\rm e}^{-ia}}{\it Ei} \left ( 1,-ib{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{ia}}{\it Ei} \left ( 1,-ib{x}^{2} \right ) }{16}}-{\frac{{{\rm e}^{3\,ia}}{\it Ei} \left ( 1,-3\,ib{x}^{2} \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.47111, size = 120, normalized size = 2.18 \begin{align*} \frac{1}{16} \,{\left ({\rm Ei}\left (3 i \, b x^{2}\right ) +{\rm Ei}\left (-3 i \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + \frac{3}{16} \,{\left ({\rm Ei}\left (i \, b x^{2}\right ) +{\rm Ei}\left (-i \, b x^{2}\right )\right )} \cos \left (a\right ) + \frac{1}{16} \,{\left (i \,{\rm Ei}\left (3 i \, b x^{2}\right ) - i \,{\rm Ei}\left (-3 i \, b x^{2}\right )\right )} \sin \left (3 \, a\right ) + \frac{1}{16} \,{\left (3 i \,{\rm Ei}\left (i \, b x^{2}\right ) - 3 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 = 1.59335, size = 261, normalized size = 4.75 \begin{align*} \frac{1}{16} \,{\left (\operatorname{Ci}\left (3 \, b x^{2}\right ) + \operatorname{Ci}\left (-3 \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + \frac{3}{16} \,{\left (\operatorname{Ci}\left (b x^{2}\right ) + \operatorname{Ci}\left (-b x^{2}\right )\right )} \cos \left (a\right ) - \frac{1}{8} \, \sin \left (3 \, a\right ) \operatorname{Si}\left (3 \, b x^{2}\right ) - \frac{3}{8} \, \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 ^{3}{\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 = 1.16666, size = 63, normalized size = 1.15 \begin{align*} \frac{1}{8} \, \cos \left (3 \, a\right ) \operatorname{Ci}\left (3 \, b x^{2}\right ) + \frac{3}{8} \, \cos \left (a\right ) \operatorname{Ci}\left (b x^{2}\right ) - \frac{3}{8} \, \sin \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) + \frac{1}{8} \, \sin \left (3 \, a\right ) \operatorname{Si}\left (-3 \, b x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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