Optimal. Leaf size=98 \[ -\frac{i e^{i a} b x^{3/2} \text{Gamma}\left (\frac{3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac{i e^{-i a} b x^{3/2} \text{Gamma}\left (\frac{3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}}-\frac{2 \cos \left (a+b x^2\right )}{\sqrt{x}} \]
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Rubi [A] time = 0.0758502, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3388, 3389, 2218} \[ -\frac{i e^{i a} b x^{3/2} \text{Gamma}\left (\frac{3}{4},-i b x^2\right )}{\left (-i b x^2\right )^{3/4}}+\frac{i e^{-i a} b x^{3/2} \text{Gamma}\left (\frac{3}{4},i b x^2\right )}{\left (i b x^2\right )^{3/4}}-\frac{2 \cos \left (a+b x^2\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 3388
Rule 3389
Rule 2218
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b x^2\right )}{x^{3/2}} \, dx &=-\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*}
Mathematica [A] time = 0.182243, size = 114, normalized size = 1.16 \[ \frac{b x^2 \left (i b x^2\right )^{3/4} (\sin (a)-i \cos (a)) \text{Gamma}\left (\frac{3}{4},-i b x^2\right )+i \left (-i b x^2\right )^{7/4} (\sin (a)+i \cos (a)) \text{Gamma}\left (\frac{3}{4},i b x^2\right )-2 \left (b^2 x^4\right )^{3/4} \cos \left (a+b x^2\right )}{\sqrt{x} \left (b^2 x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.067, size = 338, normalized size = 3.5 \begin{align*}{\frac{\cos \left ( a \right ) \sqrt{\pi }{2}^{{\frac{3}{4}}}}{8}\sqrt [8]{{b}^{2}} \left ( -12\,{\frac{\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{\sqrt{\pi }{x}^{5/2}\sqrt [8]{{b}^{2}}b} \left ({\frac{8\,{b}^{2}{x}^{4}}{21}}+2/3 \right ) }-8\,{\frac{\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{\sqrt{\pi }{x}^{5/2}\sqrt [8]{{b}^{2}}b}}+{\frac{32\,{b}^{2}\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{7\,\sqrt{\pi }}{x}^{{\frac{7}{2}}}{\it LommelS1} \left ({\frac{5}{4}},{\frac{3}{2}},b{x}^{2} \right ){\frac{1}{\sqrt [8]{{b}^{2}}}} \left ( b{x}^{2} \right ) ^{-{\frac{5}{4}}}}+8\,{\frac{{x}^{7/2}{b}^{2}\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ){\it LommelS1} \left ( 1/4,1/2,b{x}^{2} \right ) }{\sqrt{\pi }\sqrt [8]{{b}^{2}} \left ( b{x}^{2} \right ) ^{9/4}}} \right ) }-{\frac{\sin \left ( a \right ) \sqrt{\pi }{2}^{{\frac{3}{4}}}}{8}\sqrt [4]{b} \left ({\frac{8\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{3\,\sqrt{\pi }}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt [4]{b}}}}+{\frac{32\,\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt [4]{b}}}}-{\frac{8\,\sqrt [4]{2}\sin \left ( b{x}^{2} \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{7}{2}}}{b}^{{\frac{7}{4}}}{\it LommelS1} \left ({\frac{1}{4}},{\frac{3}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{5}{4}}}}-{\frac{32\,\sqrt [4]{2} \left ( \cos \left ( b{x}^{2} \right ) b{x}^{2}-\sin \left ( b{x}^{2} \right ) \right ) }{3\,\sqrt{\pi }}{x}^{{\frac{7}{2}}}{b}^{{\frac{7}{4}}}{\it LommelS1} \left ({\frac{5}{4}},{\frac{1}{2}},b{x}^{2} \right ) \left ( b{x}^{2} \right ) ^{-{\frac{9}{4}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.41277, size = 355, normalized size = 3.62 \begin{align*} -\frac{\left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}}{\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 )}}{8 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69499, size = 163, normalized size = 1.66 \begin{align*} \frac{\left (i \, b\right )^{\frac{1}{4}} x e^{\left (-i \, a\right )} \Gamma \left (\frac{3}{4}, i \, b x^{2}\right ) + \left (-i \, b\right )^{\frac{1}{4}} x e^{\left (i \, a\right )} \Gamma \left (\frac{3}{4}, -i \, b x^{2}\right ) - 2 \, \sqrt{x} \cos \left (b x^{2} + a\right )}{x} \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 )}}{x^{\frac{3}{2}}}\, 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 )}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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