Optimal. Leaf size=81 \[ -\frac{e^{i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},-i b x^2\right )}{4 \sqrt [4]{-i b x^2}}-\frac{e^{-i a} \sqrt{x} \text{Gamma}\left (\frac{1}{4},i b x^2\right )}{4 \sqrt [4]{i b x^2}} \]
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Rubi [A] time = 0.0653032, antiderivative size = 81, normalized size of antiderivative = 1., 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 3390
Rule 2218
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*}
Mathematica [A] time = 0.0714116, size = 89, normalized size = 1.1 \[ -\frac{\sqrt{x} \left (\sqrt [4]{-i b x^2} (\cos (a)-i \sin (a)) \text{Gamma}\left (\frac{1}{4},i b x^2\right )+\sqrt [4]{i b x^2} (\cos (a)+i \sin (a)) \text{Gamma}\left (\frac{1}{4},-i b x^2\right )\right )}{4 \sqrt [4]{b^2 x^4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.078, size = 338, normalized size = 4.2 \begin{align*}{\frac{\cos \left ( a \right ) \sqrt{\pi }\sqrt [4]{2}}{4} \left ( 6\,{\frac{{2}^{3/4}\sqrt [8]{{b}^{2}}\sin \left ( b{x}^{2} \right ) }{\sqrt{\pi }{x}^{3/2}b} \left ({\frac{8\,{b}^{2}{x}^{4}}{27}}+2/3 \right ) }+4\,{\frac{{2}^{3/4}\sqrt [8]{{b}^{2}} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{\sqrt{\pi }{x}^{3/2}b}}-{\frac{16\,{b}^{2}{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{9\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}\sqrt [8]{{b}^{2}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}-4\,{\frac{{x}^{9/2}\sqrt [8]{{b}^{2}}{b}^{2}{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ){\it LommelS1} \left ( 3/4,1/2,b{x}^{2} \right ) }{\sqrt{\pi } \left ( b{x}^{2} \right ) ^{11/4}}} \right ){\frac{1}{\sqrt [8]{{b}^{2}}}}}-{\frac{\sin \left ( a \right ) \sqrt{\pi }\sqrt [4]{2}}{4} \left ({\frac{4\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{5\,\sqrt{\pi }}\sqrt{x}\sqrt [4]{b}}-{\frac{16\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{5\,\sqrt{\pi }}\sqrt{x}\sqrt [4]{b}}-{\frac{4\,{2}^{3/4}\sin \left ( b{x}^{2} \right ) }{5\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{b}^{{\frac{9}{4}}}{\it LommelS1} \left ({\frac{3}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{7}{4}}}}+{\frac{16\,{2}^{3/4} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{5\,\sqrt{\pi }}{x}^{{\frac{9}{2}}}{b}^{{\frac{9}{4}}}{\it LommelS1} \left ({\frac{7}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{11}{4}}}} \right ){\frac{1}{\sqrt [4]{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.44696, size = 359, normalized size = 4.43 \begin{align*} -\frac{{\left ({\left ({\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (-i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left ({\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (\frac{1}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{1}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{1}{8} \, \pi + \frac{1}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \sqrt{x}}{8 \, \left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71051, size = 132, normalized size = 1.63 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b x^{2} \right )}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x^{2} + a\right )}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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