Optimal. Leaf size=97 \[ -\frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x} \]
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Rubi [A] time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3409, 3385, 3354, 3352, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{x}\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3385
Rule 3409
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}\\ &=\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\cos (a) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}+\frac {\sin (a) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{x}\right )}{2 b}\\ &=\frac {\cos \left (a+\frac {b}{x^2}\right )}{2 b x}-\frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right ) \sin (a)}{2 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 89, normalized size = 0.92 \[ \frac {-\sqrt {2 \pi } x \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+\sqrt {2 \pi } x \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{x}\right )+2 \sqrt {b} \cos \left (a+\frac {b}{x^2}\right )}{4 b^{3/2} x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 85, normalized size = 0.88 \[ -\frac {\sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) - \sqrt {2} \pi x \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{x}\right ) \sin \relax (a) - 2 \, b \cos \left (\frac {a x^{2} + b}{x^{2}}\right )}{4 \, b^{2} x} \]
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 \[ \int \frac {\sin \left (a + \frac {b}{x^{2}}\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 65, normalized size = 0.67 \[ \frac {\cos \left (a +\frac {b}{x^{2}}\right )}{2 b x}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, x}\right )\right )}{4 b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.40, size = 74, normalized size = 0.76 \[ -\frac {\sqrt {2} {\left (x^{4}\right )}^{\frac {3}{2}} {\left ({\left (\left (i - 1\right ) \, \Gamma \left (\frac {3}{2}, \frac {i \, b}{x^{2}}\right ) - \left (i + 1\right ) \, \Gamma \left (\frac {3}{2}, -\frac {i \, b}{x^{2}}\right )\right )} \cos \relax (a) + {\left (\left (i + 1\right ) \, \Gamma \left (\frac {3}{2}, \frac {i \, b}{x^{2}}\right ) - \left (i - 1\right ) \, \Gamma \left (\frac {3}{2}, -\frac {i \, b}{x^{2}}\right )\right )} \sin \relax (a)\right )} \left (\frac {b^{2}}{x^{4}}\right )^{\frac {3}{4}}}{8 \, b^{3} x^{3}} \]
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 \[ \int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^4} \,d x \]
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 \[ \int \frac {\sin {\left (a + \frac {b}{x^{2}} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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