Optimal. Leaf size=80 \[ -\sqrt{2 \pi } \sqrt{b} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )-\sqrt{2 \pi } \sqrt{b} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )-\frac{\cos \left (a+b x^2\right )}{x} \]
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Rubi [A] time = 0.0445045, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3388, 3353, 3352, 3351} \[ -\sqrt{2 \pi } \sqrt{b} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )-\sqrt{2 \pi } \sqrt{b} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )-\frac{\cos \left (a+b x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 3388
Rule 3353
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b x^2\right )}{x^2} \, dx &=-\frac{\cos \left (a+b x^2\right )}{x}-(2 b) \int \sin \left (a+b x^2\right ) \, dx\\ &=-\frac{\cos \left (a+b x^2\right )}{x}-(2 b \cos (a)) \int \sin \left (b x^2\right ) \, dx-(2 b \sin (a)) \int \cos \left (b x^2\right ) \, dx\\ &=-\frac{\cos \left (a+b x^2\right )}{x}-\sqrt{b} \sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )-\sqrt{b} \sqrt{2 \pi } C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sin (a)\\ \end{align*}
Mathematica [A] time = 0.201662, size = 81, normalized size = 1.01 \[ -\sqrt{2 \pi } \sqrt{b} \left (\sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )+\cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )\right )+\frac{\sin (a) \sin \left (b x^2\right )}{x}-\frac{\cos (a) \cos \left (b x^2\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 57, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( b{x}^{2}+a \right ) }{x}}-\sqrt{b}\sqrt{2}\sqrt{\pi } \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.73907, size = 355, normalized size = 4.44 \begin{align*} -\frac{\sqrt{x^{2}{\left | b \right |}}{\left ({\left ({\left (\Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) - i \, \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) +{\left ({\left (-i \, \Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + i \, \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (-\frac{1}{2}, i \, b x^{2}\right ) + \Gamma \left (-\frac{1}{2}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )}}{8 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68656, size = 209, normalized size = 2.61 \begin{align*} -\frac{\sqrt{2} \pi x \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) + \sqrt{2} \pi x \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) + \cos \left (b x^{2} + a\right )}{x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{2} + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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