Optimal. Leaf size=99 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]
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Rubi [A] time = 0.113156, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {3416, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )}{b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3416
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b \sqrt [3]{x}\right )}{\sqrt{x}} \, dx &=3 \operatorname{Subst}\left (\int \sqrt{x} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{(3 \cos (a)) \operatorname{Subst}\left (\int \frac{\sin (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{2 b}-\frac{(3 \sin (a)) \operatorname{Subst}\left (\int \frac{\cos (b x)}{\sqrt{x}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac{(3 \cos (a)) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{b}-\frac{(3 \sin (a)) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{b}\\ &=-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )}{b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{b^{3/2}}+\frac{3 \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.152974, size = 94, normalized size = 0.95 \[ -\frac{3 \left (\sqrt{2 \pi } \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [6]{x}\right )+\sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [6]{x}\right )-2 \sqrt{b} \sqrt [6]{x} \sin \left (a+b \sqrt [3]{x}\right )\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 64, normalized size = 0.7 \begin{align*} 3\,{\frac{\sqrt [6]{x}\sin \left ( a+b\sqrt [3]{x} \right ) }{b}}-{\frac{3\,\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt [6]{x}\sqrt{b}} \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 = 1.95197, size = 346, normalized size = 3.49 \begin{align*} \frac{3 \,{\left (8 \, x^{\frac{1}{6}}{\left | b \right |} \sin \left (b x^{\frac{1}{3}} + a\right ) + \sqrt{\pi }{\left ({\left ({\left (-i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \operatorname{erf}\left (\sqrt{i \, b} x^{\frac{1}{6}}\right ) +{\left ({\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )\right )} \operatorname{erf}\left (\sqrt{-i \, b} x^{\frac{1}{6}}\right )\right )} \sqrt{{\left | b \right |}}\right )}}{8 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92261, size = 250, normalized size = 2.53 \begin{align*} -\frac{3 \,{\left (\sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{S}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) + \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\sqrt{2} x^{\frac{1}{6}} \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) - 2 \, b x^{\frac{1}{6}} \sin \left (b x^{\frac{1}{3}} + a\right )\right )}}{2 \, 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{\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 = 1.16036, size = 193, normalized size = 1.95 \begin{align*} -\frac{3 i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x^{\frac{1}{6}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{4 \, b{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} + \frac{3 i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x^{\frac{1}{6}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{4 \, b{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} - \frac{3 i \, x^{\frac{1}{6}} e^{\left (i \, b x^{\frac{1}{3}} + i \, a\right )}}{2 \, b} + \frac{3 i \, x^{\frac{1}{6}} e^{\left (-i \, b x^{\frac{1}{3}} - i \, a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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