Optimal. Leaf size=75 \[ -\frac {\sqrt {\frac {\pi }{2}} \sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 75, 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 = {3383, 3353, 3352, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} \sin (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{x}\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3383
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^2} \, dx &=-\operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\left (\cos (a) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )\right )-\sin (a) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sqrt {\frac {\pi }{2}} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 61, normalized size = 0.81 \[ -\frac {\sqrt {\frac {\pi }{2}} \left (\sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 64, normalized size = 0.85 \[ -\frac {\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) + \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) \sin \relax (a)}{2 \, b} \]
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 \[ \int \frac {\sin \left (a + \frac {b}{x^{2}}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 47, normalized size = 0.63 \[ -\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )+\sin \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{2 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 98, normalized size = 1.31 \[ -\frac {\sqrt {2} \sqrt {x^{4}} {\left ({\left (\left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \cos \relax (a) + {\left (-\left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \sin \relax (a)\right )} \left (\frac {b^{2}}{x^{4}}\right )^{\frac {1}{4}}}{8 \, b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.80, size = 55, normalized size = 0.73 \[ -\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}}{x\,\sqrt {\pi }}\right )\,\cos \relax (a)}{2\,\sqrt {b}}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}}{x\,\sqrt {\pi }}\right )\,\sin \relax (a)}{2\,\sqrt {b}} \]
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 \[ \int \frac {\sin {\left (a + \frac {b}{x^{2}} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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