Optimal. Leaf size=42 \[ -\frac {1}{2} b \sin (a) \text {Ci}\left (b x^2\right )-\frac {1}{2} b \cos (a) \text {Si}\left (b x^2\right )-\frac {\cos \left (a+b x^2\right )}{2 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 42, 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 = {3380, 3297, 3303, 3299, 3302} \[ -\frac {1}{2} b \sin (a) \text {CosIntegral}\left (b x^2\right )-\frac {1}{2} b \cos (a) \text {Si}\left (b x^2\right )-\frac {\cos \left (a+b x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3380
Rubi steps
\begin {align*} \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\cos \left (a+b x^2\right )}{2 x^2}-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {\cos \left (a+b x^2\right )}{2 x^2}-\frac {1}{2} (b \cos (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^2\right )-\frac {1}{2} (b \sin (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {\cos \left (a+b x^2\right )}{2 x^2}-\frac {1}{2} b \text {Ci}\left (b x^2\right ) \sin (a)-\frac {1}{2} b \cos (a) \text {Si}\left (b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 42, normalized size = 1.00 \[ -\frac {b x^2 \sin (a) \text {Ci}\left (b x^2\right )+b x^2 \cos (a) \text {Si}\left (b x^2\right )+\cos \left (a+b x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 57, normalized size = 1.36 \[ -\frac {2 \, b x^{2} \cos \relax (a) \operatorname {Si}\left (b x^{2}\right ) + {\left (b x^{2} \operatorname {Ci}\left (b x^{2}\right ) + b x^{2} \operatorname {Ci}\left (-b x^{2}\right )\right )} \sin \relax (a) + 2 \, \cos \left (b x^{2} + a\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 87, normalized size = 2.07 \[ -\frac {{\left (b x^{2} + a\right )} b^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \relax (a) - a b^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \relax (a) + {\left (b x^{2} + a\right )} b^{2} \cos \relax (a) \operatorname {Si}\left (b x^{2}\right ) - a b^{2} \cos \relax (a) \operatorname {Si}\left (b x^{2}\right ) + b^{2} \cos \left (b x^{2} + a\right )}{2 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 39, normalized size = 0.93 \[ -\frac {\cos \left (b \,x^{2}+a \right )}{2 x^{2}}-b \left (\frac {\cos \relax (a ) \Si \left (b \,x^{2}\right )}{2}+\frac {\sin \relax (a ) \Ci \left (b \,x^{2}\right )}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.07, size = 48, normalized size = 1.14 \[ -\frac {1}{4} \, {\left ({\left (i \, \Gamma \left (-1, i \, b x^{2}\right ) - i \, \Gamma \left (-1, -i \, b x^{2}\right )\right )} \cos \relax (a) + {\left (\Gamma \left (-1, i \, b x^{2}\right ) + \Gamma \left (-1, -i \, b x^{2}\right )\right )} \sin \relax (a)\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (b\,x^2+a\right )}{x^3} \,d x \]
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 \[ \int \frac {\cos {\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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