Optimal. Leaf size=81 \[ -\frac {e^{i a} \sqrt {x} \Gamma \left (\frac {1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac {e^{-i a} \sqrt {x} \Gamma \left (\frac {1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3390, 2218} \[ -\frac {e^{i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac {e^{-i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3390
Rubi steps
\begin {align*} \int \frac {\cos \left (a+b x^2\right )}{\sqrt {x}} \, dx &=\frac {1}{2} \int \frac {e^{-i a-i b x^2}}{\sqrt {x}} \, dx+\frac {1}{2} \int \frac {e^{i a+i b x^2}}{\sqrt {x}} \, dx\\ &=-\frac {e^{i a} \sqrt {x} \Gamma \left (\frac {1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac {e^{-i a} \sqrt {x} \Gamma \left (\frac {1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 89, normalized size = 1.10 \[ -\frac {\sqrt {x} \left (\sqrt [4]{-i b x^2} (\cos (a)-i \sin (a)) \Gamma \left (\frac {1}{4},i b x^2\right )+\sqrt [4]{i b x^2} (\cos (a)+i \sin (a)) \Gamma \left (\frac {1}{4},-i b x^2\right )\right )}{4 \sqrt [4]{b^2 x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 44, normalized size = 0.54 \[ \frac {i \, \left (i \, b\right )^{\frac {3}{4}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) - i \, \left (-i \, b\right )^{\frac {3}{4}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )}{4 \, b} \]
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 \[ \int \frac {\cos \left (b x^{2} + a\right )}{\sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 338, normalized size = 4.17 \[ \frac {\cos \relax (a ) \sqrt {\pi }\, 2^{\frac {1}{4}} \left (\frac {6 \,2^{\frac {3}{4}} \left (b^{2}\right )^{\frac {1}{8}} \left (\frac {8 x^{4} b^{2}}{27}+\frac {2}{3}\right ) \sin \left (b \,x^{2}\right )}{\sqrt {\pi }\, x^{\frac {3}{2}} b}+\frac {4 \,2^{\frac {3}{4}} \left (b^{2}\right )^{\frac {1}{8}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{\sqrt {\pi }\, x^{\frac {3}{2}} b}-\frac {16 x^{\frac {9}{2}} \left (b^{2}\right )^{\frac {1}{8}} b^{2} 2^{\frac {3}{4}} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {7}{4}, \frac {3}{2}, b \,x^{2}\right )}{9 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {7}{4}}}-\frac {4 x^{\frac {9}{2}} \left (b^{2}\right )^{\frac {1}{8}} b^{2} 2^{\frac {3}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {3}{4}, \frac {1}{2}, b \,x^{2}\right )}{\sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {11}{4}}}\right )}{4 \left (b^{2}\right )^{\frac {1}{8}}}-\frac {\sin \relax (a ) \sqrt {\pi }\, 2^{\frac {1}{4}} \left (\frac {4 \sqrt {x}\, 2^{\frac {3}{4}} b^{\frac {1}{4}} \sin \left (b \,x^{2}\right )}{5 \sqrt {\pi }}-\frac {16 \sqrt {x}\, 2^{\frac {3}{4}} b^{\frac {1}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right )}{5 \sqrt {\pi }}-\frac {4 x^{\frac {9}{2}} b^{\frac {9}{4}} 2^{\frac {3}{4}} \sin \left (b \,x^{2}\right ) \LommelS 1 \left (\frac {3}{4}, \frac {3}{2}, b \,x^{2}\right )}{5 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {7}{4}}}+\frac {16 x^{\frac {9}{2}} b^{\frac {9}{4}} 2^{\frac {3}{4}} \left (\cos \left (b \,x^{2}\right ) x^{2} b -\sin \left (b \,x^{2}\right )\right ) \LommelS 1 \left (\frac {7}{4}, \frac {1}{2}, b \,x^{2}\right )}{5 \sqrt {\pi }\, \left (b \,x^{2}\right )^{\frac {11}{4}}}\right )}{4 b^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.84, size = 135, normalized size = 1.67 \[ -\frac {{\left ({\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )} - \sqrt {-\sqrt {2} + 2} {\left (i \, \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )}\right )} \cos \relax (a) - {\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )} + \sqrt {\sqrt {2} + 2} {\left (i \, \Gamma \left (\frac {1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac {1}{4}, -i \, b x^{2}\right )\right )}\right )} \sin \relax (a)\right )} \sqrt {x}}{8 \, \left (b x^{2}\right )^{\frac {1}{4}}} \]
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 \[ \int \frac {\cos \left (b\,x^2+a\right )}{\sqrt {x}} \,d x \]
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 \[ \int \frac {\cos {\left (a + b x^{2} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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