Integrand size = 14, antiderivative size = 98 \[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=-\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-\frac {i b e^{i a} x^{3/2} \Gamma \left (\frac {3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac {i b e^{-i a} x^{3/2} \Gamma \left (\frac {3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}} \]
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Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3469, 3470, 2250} \[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=-\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-\frac {i e^{i a} b x^{3/2} \Gamma \left (\frac {3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac {i e^{-i a} b x^{3/2} \Gamma \left (\frac {3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}} \]
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Rule 2250
Rule 3469
Rule 3470
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-(4 b) \int \sqrt {x} \sin \left (a+b x^2\right ) \, dx \\ & = -\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-(2 i b) \int e^{-i a-i b x^2} \sqrt {x} \, dx+(2 i b) \int e^{i a+i b x^2} \sqrt {x} \, dx \\ & = -\frac {2 \cos \left (a+b x^2\right )}{\sqrt {x}}-\frac {i b e^{i a} x^{3/2} \Gamma \left (\frac {3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac {i b e^{-i a} x^{3/2} \Gamma \left (\frac {3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16 \[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=\frac {-2 \left (b^2 x^4\right )^{3/4} \cos \left (a+b x^2\right )+b x^2 \left (i b x^2\right )^{3/4} \Gamma \left (\frac {3}{4},-i b x^2\right ) (-i \cos (a)+\sin (a))+i \left (-i b x^2\right )^{7/4} \Gamma \left (\frac {3}{4},i b x^2\right ) (i \cos (a)+\sin (a))}{\sqrt {x} \left (b^2 x^4\right )^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.43 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.45
method | result | size |
meijerg | \(\frac {\cos \left (a \right ) \sqrt {\pi }\, 2^{\frac {3}{4}} \left (b^{2}\right )^{\frac {1}{8}} \left (-\frac {12 \,2^{\frac {1}{4}} \left (\frac {8 x^{4} b^{2}}{21}+\frac {2}{3}\right ) \sin \left (b \,x^{2}\right )}{\sqrt {\pi }\, x^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{8}} b}-\frac {8 \,2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{\sqrt {\pi }\, x^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{8}} b}+\frac {32 x^{\frac {7}{2}} b^{2} 2^{\frac {1}{4}} \sin \left (b \,x^{2}\right ) s_{\frac {5}{4},\frac {3}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{7 \sqrt {\pi }\, \left (b^{2}\right )^{\frac {1}{8}} \left (b \,x^{2}\right )^{\frac {5}{4}}}+\frac {8 x^{\frac {7}{2}} b^{2} 2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) s_{\frac {1}{4},\frac {1}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{\sqrt {\pi }\, \left (b^{2}\right )^{\frac {1}{8}} \left (b \,x^{2}\right )^{\frac {9}{4}}}\right )}{8}-\frac {\sin \left (a \right ) \sqrt {\pi }\, 2^{\frac {3}{4}} b^{\frac {1}{4}} \left (\frac {8 \,2^{\frac {1}{4}} \sin \left (b \,x^{2}\right )}{3 \sqrt {\pi }\, \sqrt {x}\, b^{\frac {1}{4}}}+\frac {32 \,2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{3 \sqrt {\pi }\, \sqrt {x}\, b^{\frac {1}{4}}}-\frac {8 x^{\frac {7}{2}} b^{\frac {7}{4}} 2^{\frac {1}{4}} \sin \left (b \,x^{2}\right ) s_{\frac {1}{4},\frac {3}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{3 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {5}{4}}}-\frac {32 x^{\frac {7}{2}} b^{\frac {7}{4}} 2^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) s_{\frac {5}{4},\frac {1}{2}}^{\left (+\right )}\left (b \,x^{2}\right )}{3 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {9}{4}}}\right )}{8}\) | \(338\) |
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Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.67 \[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=\frac {{\left (x \cos \left (a\right ) - i \, x \sin \left (a\right )\right )} \left (i \, b\right )^{\frac {1}{4}} \Gamma \left (\frac {3}{4}, i \, b x^{2}\right ) + {\left (x \cos \left (a\right ) + i \, x \sin \left (a\right )\right )} \left (-i \, b\right )^{\frac {1}{4}} \Gamma \left (\frac {3}{4}, -i \, b x^{2}\right ) - 2 \, \sqrt {x} \cos \left (b x^{2} + a\right )}{x} \]
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\[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=\int \frac {\cos {\left (a + b x^{2} \right )}}{x^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=\int { \frac {\cos \left (b x^{2} + a\right )}{x^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos \left (a+b x^2\right )}{x^{3/2}} \, dx=\int \frac {\cos \left (b\,x^2+a\right )}{x^{3/2}} \,d x \]
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