Optimal. Leaf size=100 \[ -\frac{e^{2 i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},-2 i b x^2\right )}{8\ 2^{3/4} \left (-i b x^2\right )^{3/4}}-\frac{e^{-2 i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},2 i b x^2\right )}{8\ 2^{3/4} \left (i b x^2\right )^{3/4}}+\frac{x^{3/2}}{3} \]
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Rubi [A] time = 0.130385, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3402, 3404, 3390, 2218} \[ -\frac{e^{2 i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},-2 i b x^2\right )}{8\ 2^{3/4} \left (-i b x^2\right )^{3/4}}-\frac{e^{-2 i a} x^{3/2} \text{Gamma}\left (\frac{3}{4},2 i b x^2\right )}{8\ 2^{3/4} \left (i b x^2\right )^{3/4}}+\frac{x^{3/2}}{3} \]
Antiderivative was successfully verified.
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Rule 3402
Rule 3404
Rule 3390
Rule 2218
Rubi steps
\begin{align*} \int \sqrt{x} \cos ^2\left (a+b x^2\right ) \, dx &=2 \operatorname{Subst}\left (\int x^2 \cos ^2\left (a+b x^4\right ) \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^2}{2}+\frac{1}{2} x^2 \cos \left (2 a+2 b x^4\right )\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{3/2}}{3}+\operatorname{Subst}\left (\int x^2 \cos \left (2 a+2 b x^4\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{3/2}}{3}+\frac{1}{2} \operatorname{Subst}\left (\int e^{-2 i a-2 i b x^4} x^2 \, dx,x,\sqrt{x}\right )+\frac{1}{2} \operatorname{Subst}\left (\int e^{2 i a+2 i b x^4} x^2 \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{3/2}}{3}-\frac{e^{2 i a} x^{3/2} \Gamma \left (\frac{3}{4},-2 i b x^2\right )}{8\ 2^{3/4} \left (-i b x^2\right )^{3/4}}-\frac{e^{-2 i a} x^{3/2} \Gamma \left (\frac{3}{4},2 i b x^2\right )}{8\ 2^{3/4} \left (i b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.226213, size = 122, normalized size = 1.22 \[ \frac{x^{3/2} \left (-3 \sqrt [4]{2} \left (-i b x^2\right )^{3/4} (\cos (2 a)-i \sin (2 a)) \text{Gamma}\left (\frac{3}{4},2 i b x^2\right )-3 \sqrt [4]{2} \left (i b x^2\right )^{3/4} (\cos (2 a)+i \sin (2 a)) \text{Gamma}\left (\frac{3}{4},-2 i b x^2\right )+16 \left (b^2 x^4\right )^{3/4}\right )}{48 \left (b^2 x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \left ( \cos \left ( b{x}^{2}+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4219, size = 392, normalized size = 3.92 \begin{align*} \frac{32 \, x^{2}{\left | b \right |} - 2^{\frac{1}{4}} \left (x^{2}{\left | b \right |}\right )^{\frac{1}{4}}{\left ({\left (3 \,{\left (\Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) + 3 \,{\left (\Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (3 i \, \Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) - 3 i \, \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) -{\left (-3 i \, \Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) + 3 i \, \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \cos \left (2 \, a\right ) -{\left ({\left (3 i \, \Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) - 3 i \, \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \cos \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) +{\left (3 i \, \Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) - 3 i \, \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \cos \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) + 3 \,{\left (\Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \sin \left (\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right ) - 3 \,{\left (\Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )\right )} \sin \left (-\frac{3}{8} \, \pi + \frac{3}{4} \, \arctan \left (0, b\right )\right )\right )} \sin \left (2 \, a\right )\right )}}{96 \, \sqrt{x}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7438, size = 176, normalized size = 1.76 \begin{align*} \frac{16 \, b x^{\frac{3}{2}} + 3 i \, \left (2 i \, b\right )^{\frac{1}{4}} e^{\left (-2 i \, a\right )} \Gamma \left (\frac{3}{4}, 2 i \, b x^{2}\right ) - 3 i \, \left (-2 i \, b\right )^{\frac{1}{4}} e^{\left (2 i \, a\right )} \Gamma \left (\frac{3}{4}, -2 i \, b x^{2}\right )}{48 \, 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 ^{2}{\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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