Integrand size = 14, antiderivative size = 76 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=-\frac {\cos ^2\left (a+b x^2\right )}{x}-\sqrt {b} \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )-\sqrt {b} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a) \]
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Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3475, 4669, 3454, 3434, 3433, 3432} \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=-\sqrt {\pi } \sqrt {b} \sin (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )-\sqrt {\pi } \sqrt {b} \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )-\frac {\cos ^2\left (a+b x^2\right )}{x} \]
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Rule 3432
Rule 3433
Rule 3434
Rule 3454
Rule 3475
Rule 4669
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^2\left (a+b x^2\right )}{x}-(4 b) \int \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right ) \, dx \\ & = -\frac {\cos ^2\left (a+b x^2\right )}{x}-(2 b) \int \sin \left (2 \left (a+b x^2\right )\right ) \, dx \\ & = -\frac {\cos ^2\left (a+b x^2\right )}{x}-(2 b) \int \sin \left (2 a+2 b x^2\right ) \, dx \\ & = -\frac {\cos ^2\left (a+b x^2\right )}{x}-(2 b \cos (2 a)) \int \sin \left (2 b x^2\right ) \, dx-(2 b \sin (2 a)) \int \cos \left (2 b x^2\right ) \, dx \\ & = -\frac {\cos ^2\left (a+b x^2\right )}{x}-\sqrt {b} \sqrt {\pi } \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )-\sqrt {b} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=-\frac {\cos ^2\left (a+b x^2\right )+\sqrt {b} \sqrt {\pi } x \cos (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right )+\sqrt {b} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {2 \sqrt {b} x}{\sqrt {\pi }}\right ) \sin (2 a)}{x} \]
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Time = 0.40 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {1}{2 x}-\frac {\cos \left (2 b \,x^{2}+2 a \right )}{2 x}-\sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {S}\left (\frac {2 x \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \operatorname {C}\left (\frac {2 x \sqrt {b}}{\sqrt {\pi }}\right )\right )\) | \(62\) |
risch | \(-\frac {1}{2 x}-\frac {i {\mathrm e}^{-2 i a} b \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {i b}\, x \right )}{4 \sqrt {i b}}+\frac {i {\mathrm e}^{2 i a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-2 i b}\, x \right )}{2 \sqrt {-2 i b}}-\frac {\cos \left (2 b \,x^{2}+2 a \right )}{2 x}\) | \(83\) |
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none
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=-\frac {\pi x \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x \sqrt {\frac {b}{\pi }}\right ) + \pi x \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) + \cos \left (b x^{2} + a\right )^{2}}{x} \]
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\[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=\int \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=\frac {\sqrt {2} \sqrt {b x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, 2 i \, b x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, 2 i \, b x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right )\right )} - 8}{16 \, x} \]
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\[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=\int { \frac {\cos \left (b x^{2} + a\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^2} \, dx=\int \frac {{\cos \left (b\,x^2+a\right )}^2}{x^2} \,d x \]
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