Optimal. Leaf size=102 \[ -\frac{3 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{8 b^{3/2}}-\frac{3 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{8 b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}+\sqrt{x} \]
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Rubi [A] time = 0.169637, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3416, 3312, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{3 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{8 b^{3/2}}-\frac{3 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{8 b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}+\sqrt{x} \]
Antiderivative was successfully verified.
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Rule 3416
Rule 3312
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt{x}} \, dx &=3 \operatorname{Subst}\left (\int \sqrt{x} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{\sqrt{x}}{2}+\frac{1}{2} \sqrt{x} \cos (2 a+2 b x)\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\sqrt{x}+\frac{3}{2} \operatorname{Subst}\left (\int \sqrt{x} \cos (2 a+2 b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\sqrt{x}+\frac{3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 a+2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{8 b}\\ &=\sqrt{x}+\frac{3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}-\frac{(3 \cos (2 a)) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{8 b}-\frac{(3 \sin (2 a)) \operatorname{Subst}\left (\int \frac{\cos (2 b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{8 b}\\ &=\sqrt{x}+\frac{3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}-\frac{(3 \cos (2 a)) \operatorname{Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{4 b}-\frac{(3 \sin (2 a)) \operatorname{Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{4 b}\\ &=\sqrt{x}-\frac{3 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )}{8 b^{3/2}}-\frac{3 \sqrt{\pi } C\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right ) \sin (2 a)}{8 b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.189478, size = 103, normalized size = 1.01 \[ \frac{-3 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )-3 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} \sqrt [6]{x}}{\sqrt{\pi }}\right )+2 \sqrt{b} \sqrt [6]{x} \left (3 \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4 b \sqrt [3]{x}\right )}{8 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 67, normalized size = 0.7 \begin{align*} \sqrt{x}+{\frac{3}{4\,b}\sqrt [6]{x}\sin \left ( 2\,a+2\,b\sqrt [3]{x} \right ) }-{\frac{3\,\sqrt{\pi }}{8} \left ( \cos \left ( 2\,a \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) +\sin \left ( 2\,a \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt [6]{x}\sqrt{b}}{\sqrt{\pi }}} \right ) \right ){b}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.96517, size = 392, normalized size = 3.84 \begin{align*} \frac{\sqrt{2} \sqrt{\pi }{\left ({\left ({\left (-3 i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - 3 i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - 3 \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (2 \, a\right ) -{\left (3 \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - 3 i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (2 \, a\right )\right )} \operatorname{erf}\left (\sqrt{2 i \, b} x^{\frac{1}{6}}\right ) +{\left ({\left (3 i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - 3 \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (2 \, a\right ) -{\left (3 \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + 3 i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - 3 i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (2 \, a\right )\right )} \operatorname{erf}\left (\sqrt{-2 i \, b} x^{\frac{1}{6}}\right )\right )} \sqrt{{\left | b \right |}} + 64 \, b \sqrt{x}{\left | b \right |} + 48 \, x^{\frac{1}{6}}{\left | b \right |} \sin \left (2 \, b x^{\frac{1}{3}} + 2 \, a\right )}{64 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08258, size = 271, normalized size = 2.66 \begin{align*} -\frac{3 \, \pi \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{S}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) + 3 \, \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (2 \, x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) - 12 \, b x^{\frac{1}{6}} \cos \left (b x^{\frac{1}{3}} + a\right ) \sin \left (b x^{\frac{1}{3}} + a\right ) - 8 \, b^{2} \sqrt{x}}{8 \, b^{2}} \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 ^{2}{\left (a + b \sqrt [3]{x} \right )}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.17301, size = 167, normalized size = 1.64 \begin{align*} \sqrt{x} - \frac{3 i \, x^{\frac{1}{6}} e^{\left (2 i \, b x^{\frac{1}{3}} + 2 i \, a\right )}}{8 \, b} + \frac{3 i \, x^{\frac{1}{6}} e^{\left (-2 i \, b x^{\frac{1}{3}} - 2 i \, a\right )}}{8 \, b} - \frac{3 i \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b} x^{\frac{1}{6}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{16 \, b^{\frac{3}{2}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}} + \frac{3 i \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b} x^{\frac{1}{6}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{16 \, b^{\frac{3}{2}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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