Optimal. Leaf size=80 \[ \sqrt {2 \pi } \left (-\sqrt {b}\right ) \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+x \sin \left (a+\frac {b}{x^2}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3359, 3387, 3354, 3352, 3351} \[ \sqrt {2 \pi } \left (-\sqrt {b}\right ) \cos (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{x}\right )+\sqrt {2 \pi } \sqrt {b} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+x \sin \left (a+\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3359
Rule 3387
Rubi steps
\begin {align*} \int \sin \left (a+\frac {b}{x^2}\right ) \, dx &=-\operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=x \sin \left (a+\frac {b}{x^2}\right )-(2 b) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=x \sin \left (a+\frac {b}{x^2}\right )-(2 b \cos (a)) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )+(2 b \sin (a)) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\sqrt {b} \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\sqrt {b} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)+x \sin \left (a+\frac {b}{x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 81, normalized size = 1.01 \[ -\sqrt {2 \pi } \sqrt {b} \left (\cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )-\sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )\right )+x \sin (a) \cos \left (\frac {b}{x^2}\right )+x \cos (a) \sin \left (\frac {b}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 74, normalized size = 0.92 \[ -\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) + \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) \sin \relax (a) + x \sin \left (\frac {a x^{2} + b}{x^{2}}\right ) \]
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 \[ \int \sin \left (a + \frac {b}{x^{2}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 59, normalized size = 0.74 \[ x \sin \left (a +\frac {b}{x^{2}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 127, normalized size = 1.59 \[ \frac {\sqrt {2} {\left (2 \, \sqrt {2} b x^{2} \sqrt {\frac {1}{x^{4}}} \sin \left (\frac {a x^{2} + b}{x^{2}}\right ) + {\left ({\left (\left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \cos \relax (a) + {\left (\left (i + 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {\frac {i \, b}{x^{2}}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {i \, b}{x^{2}}}\right ) - 1\right )}\right )} \sin \relax (a)\right )} b \left (\frac {b^{2}}{x^{4}}\right )^{\frac {1}{4}}\right )} \sqrt {x^{4}}}{4 \, b x} \]
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 \[ \int \sin \left (a+\frac {b}{x^2}\right ) \,d x \]
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 \[ \int \sin {\left (a + \frac {b}{x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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