Integrand size = 12, antiderivative size = 42 \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=-\frac {\cos \left (a+b x^2\right )}{2 x^2}-\frac {1}{2} b \operatorname {CosIntegral}\left (b x^2\right ) \sin (a)-\frac {1}{2} b \cos (a) \text {Si}\left (b x^2\right ) \]
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Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3461, 3378, 3384, 3380, 3383} \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=-\frac {1}{2} b \sin (a) \operatorname {CosIntegral}\left (b x^2\right )-\frac {1}{2} b \cos (a) \text {Si}\left (b x^2\right )-\frac {\cos \left (a+b x^2\right )}{2 x^2} \]
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3461
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {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 \text {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)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^2\right )-\frac {1}{2} (b \sin (a)) \text {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 \operatorname {CosIntegral}\left (b x^2\right ) \sin (a)-\frac {1}{2} b \cos (a) \text {Si}\left (b x^2\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=-\frac {\cos \left (a+b x^2\right )+b x^2 \operatorname {CosIntegral}\left (b x^2\right ) \sin (a)+b x^2 \cos (a) \text {Si}\left (b x^2\right )}{2 x^2} \]
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Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {\cos \left (b \,x^{2}+a \right )}{2 x^{2}}-b \left (\frac {\cos \left (a \right ) \operatorname {Si}\left (b \,x^{2}\right )}{2}+\frac {\sin \left (a \right ) \operatorname {Ci}\left (b \,x^{2}\right )}{2}\right )\) | \(39\) |
risch | \(\frac {{\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{2}\right ) b}{4}-\frac {{\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{2}\right ) b}{2}+\frac {i \operatorname {Ei}_{1}\left (-i b \,x^{2}\right ) {\mathrm e}^{-i a} b}{4}-\frac {i {\mathrm e}^{i a} b \,\operatorname {Ei}_{1}\left (-i b \,x^{2}\right )}{4}-\frac {\cos \left (b \,x^{2}+a \right )}{2 x^{2}}\) | \(80\) |
meijerg | \(\frac {\cos \left (a \right ) \sqrt {\pi }\, \sqrt {b^{2}}\, \left (-\frac {4 b^{2} \cos \left (x^{2} \sqrt {b^{2}}\right )}{x^{2} \left (b^{2}\right )^{\frac {3}{2}} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (x^{2} \sqrt {b^{2}}\right )}{\sqrt {\pi }}\right )}{8}-\frac {\sin \left (a \right ) \sqrt {\pi }\, b \left (\frac {4 \gamma -4+8 \ln \left (x \right )+4 \ln \left (b \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \gamma }{\sqrt {\pi }}-\frac {4 \ln \left (2\right )}{\sqrt {\pi }}-\frac {4 \ln \left (\frac {b \,x^{2}}{2}\right )}{\sqrt {\pi }}-\frac {4 \sin \left (b \,x^{2}\right )}{\sqrt {\pi }\, x^{2} b}+\frac {4 \,\operatorname {Ci}\left (b \,x^{2}\right )}{\sqrt {\pi }}\right )}{8}\) | \(141\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=-\frac {b x^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \left (a\right ) + b x^{2} \cos \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) + \cos \left (b x^{2} + a\right )}{2 \, x^{2}} \]
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\[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=\int \frac {\cos {\left (a + b x^{2} \right )}}{x^{3}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=-\frac {1}{4} \, {\left ({\left (i \, \Gamma \left (-1, i \, b x^{2}\right ) - i \, \Gamma \left (-1, -i \, b x^{2}\right )\right )} \cos \left (a\right ) + {\left (\Gamma \left (-1, i \, b x^{2}\right ) + \Gamma \left (-1, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} b \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (36) = 72\).
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.07 \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=-\frac {{\left (b x^{2} + a\right )} b^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \left (a\right ) - a b^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \left (a\right ) + {\left (b x^{2} + a\right )} b^{2} \cos \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) - a b^{2} \cos \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) + b^{2} \cos \left (b x^{2} + a\right )}{2 \, b^{2} x^{2}} \]
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Timed out. \[ \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx=\int \frac {\cos \left (b\,x^2+a\right )}{x^3} \,d x \]
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