Optimal. Leaf size=80 \[ -\sqrt {2 \pi } \sqrt {b} \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sqrt {2 \pi } \sqrt {b} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\frac {\cos \left (a+b x^2\right )}{x} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 3351
Rule 3352
Rule 3353
Rule 3388
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*}
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Mathematica [A] time = 0.18, size = 81, normalized size = 1.01 \[ -\sqrt {2 \pi } \sqrt {b} \left (\sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )\right )+\frac {\sin (a) \sin \left (b x^2\right )}{x}-\frac {\cos (a) \cos \left (b x^2\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 70, normalized size = 0.88 \[ -\frac {\sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a) + \cos \left (b x^{2} + a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x^{2} + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 0.71 \[ -\frac {\cos \left (b \,x^{2}+a \right )}{x}-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \relax (a ) \FresnelC \left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.58, size = 73, normalized size = 0.91 \[ \frac {\sqrt {b x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \cos \relax (a) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, b x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, b x^{2}\right )\right )} \sin \relax (a)\right )}}{8 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (b\,x^2+a\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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