Optimal. Leaf size=81 \[ -\frac{e^{i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac{e^{-i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0629466, 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} \[ -\frac{e^{i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac{e^{-i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 3390
Rule 2218
Rubi steps
\begin{align*} \int \sqrt{x} \cos \left (a+b x^2\right ) \, dx &=\frac{1}{2} \int e^{-i a-i b x^2} \sqrt{x} \, dx+\frac{1}{2} \int e^{i a+i b x^2} \sqrt{x} \, dx\\ &=-\frac{e^{i a} x^{3/2} \Gamma \left (\frac{3}{4},-i b x^2\right )}{4 \left (-i b x^2\right )^{3/4}}-\frac{e^{-i a} x^{3/2} \Gamma \left (\frac{3}{4},i b x^2\right )}{4 \left (i b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0898811, size = 89, normalized size = 1.1 \[ -\frac{x^{3/2} \left (\left (-i b x^2\right )^{3/4} (\cos (a)-i \sin (a)) \text{Gamma}\left (\frac{3}{4},i b x^2\right )+\left (i b x^2\right )^{3/4} (\cos (a)+i \sin (a)) \text{Gamma}\left (\frac{3}{4},-i b x^2\right )\right )}{4 \left (b^2 x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.066, size = 290, normalized size = 3.6 \begin{align*}{\frac{{2}^{{\frac{3}{4}}}\cos \left ( a \right ) \sqrt{\pi }}{4} \left ({\frac{4\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{3\,\sqrt{\pi }b} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\frac{1}{\sqrt{x}}}}+{\frac{4\,\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }b} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\frac{1}{\sqrt{x}}}}-{\frac{\sqrt [4]{2}b\sin \left ( b{x}^{2} \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{7}{2}}} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\it LommelS1} \left ({\frac{1}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{5}{4}}}}-{\frac{4\,\sqrt [4]{2}b \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{7}{2}}} \left ({b}^{2} \right ) ^{{\frac{3}{8}}}{\it LommelS1} \left ({\frac{5}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{9}{4}}}} \right ) \left ({b}^{2} \right ) ^{-{\frac{3}{8}}}}-{\frac{{2}^{{\frac{3}{4}}}\sin \left ( a \right ) \sqrt{\pi }}{4} \left ({\frac{4\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{7\,\sqrt{\pi }}{x}^{{\frac{3}{2}}}{b}^{{\frac{3}{4}}}}-{\frac{4\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{7\,\sqrt{\pi }}{x}^{{\frac{7}{2}}}{b}^{{\frac{7}{4}}}{\it LommelS1} \left ({\frac{5}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{5}{4}}}}-{\frac{\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{\sqrt{\pi }}{x}^{{\frac{7}{2}}}{b}^{{\frac{7}{4}}}{\it LommelS1} \left ({\frac{1}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{9}{4}}}} \right ){b}^{-{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6082, size = 365, normalized size = 4.51 \begin{align*} -\frac{\left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}}{\left ({\left ({\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (-i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left ({\left (i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (i \, \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )}}{8 \, \sqrt{x}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7082, size = 132, normalized size = 1.63 \begin{align*} \frac{i \, \left (i \, b\right )^{\frac{1}{4}} e^{\left (-i \, a\right )} \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) - i \, \left (-i \, b\right )^{\frac{1}{4}} e^{\left (i \, a\right )} \Gamma \left (\frac{3}{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 \sqrt{x} \cos{\left (a + b x^{2} \right )}\, 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 \sqrt{x} \cos \left (b x^{2} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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