Optimal. Leaf size=57 \[ -\frac {1}{2} b \sin (2 a) \text {Ci}\left (2 b x^2\right )-\frac {1}{2} b \cos (2 a) \text {Si}\left (2 b x^2\right )-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{4 x^2} \]
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Rubi [A] time = 0.12, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3404, 3380, 3297, 3303, 3299, 3302} \[ -\frac {1}{2} b \sin (2 a) \text {CosIntegral}\left (2 b x^2\right )-\frac {1}{2} b \cos (2 a) \text {Si}\left (2 b x^2\right )-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3380
Rule 3404
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (a+b x^2\right )}{x^3} \, dx &=\int \left (\frac {1}{2 x^3}+\frac {\cos \left (2 a+2 b x^2\right )}{2 x^3}\right ) \, dx\\ &=-\frac {1}{4 x^2}+\frac {1}{2} \int \frac {\cos \left (2 a+2 b x^2\right )}{x^3} \, dx\\ &=-\frac {1}{4 x^2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^2}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\sin (2 a+2 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^2}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{2} (b \cos (2 a)) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{x} \, dx,x,x^2\right )-\frac {1}{2} (b \sin (2 a)) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^2}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{4 x^2}-\frac {1}{2} b \text {Ci}\left (2 b x^2\right ) \sin (2 a)-\frac {1}{2} b \cos (2 a) \text {Si}\left (2 b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 50, normalized size = 0.88 \[ -\frac {b x^2 \sin (2 a) \text {Ci}\left (2 b x^2\right )+b x^2 \cos (2 a) \text {Si}\left (2 b x^2\right )+\cos ^2\left (a+b x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 65, normalized size = 1.14 \[ -\frac {2 \, b x^{2} \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{2}\right ) + 2 \, \cos \left (b x^{2} + a\right )^{2} + {\left (b x^{2} \operatorname {Ci}\left (2 \, b x^{2}\right ) + b x^{2} \operatorname {Ci}\left (-2 \, b x^{2}\right )\right )} \sin \left (2 \, a\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 107, normalized size = 1.88 \[ -\frac {2 \, {\left (b x^{2} + a\right )} b^{2} \operatorname {Ci}\left (2 \, b x^{2}\right ) \sin \left (2 \, a\right ) - 2 \, a b^{2} \operatorname {Ci}\left (2 \, b x^{2}\right ) \sin \left (2 \, a\right ) - 2 \, {\left (b x^{2} + a\right )} b^{2} \cos \left (2 \, a\right ) \operatorname {Si}\left (-2 \, b x^{2}\right ) + 2 \, a b^{2} \cos \left (2 \, a\right ) \operatorname {Si}\left (-2 \, b x^{2}\right ) + b^{2} \cos \left (2 \, b x^{2} + 2 \, a\right ) + b^{2}}{4 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 89, normalized size = 1.56 \[ -\frac {1}{4 x^{2}}+\frac {\pi \,{\mathrm e}^{-2 i a} \mathrm {csgn}\left (b \,x^{2}\right ) b}{4}-\frac {{\mathrm e}^{-2 i a} \Si \left (2 b \,x^{2}\right ) b}{2}+\frac {i {\mathrm e}^{-2 i a} \Ei \left (1, -2 i b \,x^{2}\right ) b}{4}-\frac {i {\mathrm e}^{2 i a} b \Ei \left (1, -2 i b \,x^{2}\right )}{4}-\frac {\cos \left (2 b \,x^{2}+2 a \right )}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.74, size = 61, normalized size = 1.07 \[ -\frac {{\left ({\left (i \, \Gamma \left (-1, 2 i \, b x^{2}\right ) - i \, \Gamma \left (-1, -2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + {\left (\Gamma \left (-1, 2 i \, b x^{2}\right ) + \Gamma \left (-1, -2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right )\right )} b x^{2} + 1}{4 \, x^{2}} \]
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 \[ \int \frac {{\cos \left (b\,x^2+a\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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