Optimal. Leaf size=102 \[ -\frac {3 \sqrt {\pi } \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {3 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{8 b^{3/2}}+\frac {3 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{4 b}+\sqrt {x} \]
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Rubi [A] time = 0.17, antiderivative size = 102, normalized size of antiderivative = 1.00, 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 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3351
Rule 3352
Rule 3416
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*}
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Mathematica [A] time = 0.18, size = 103, normalized size = 1.01 \[ \frac {-3 \sqrt {\pi } \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-3 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+2 \sqrt {b} \sqrt [6]{x} \left (3 \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4 b \sqrt [3]{x}\right )}{8 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 90, normalized size = 0.88 \[ -\frac {3 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 3 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 12 \, b x^{\frac {1}{6}} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) - 8 \, b^{2} \sqrt {x}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.53, size = 124, normalized size = 1.22 \[ \sqrt {x} - \frac {3 i \, x^{\frac {1}{6}} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{8 \, b} + \frac {3 i \, x^{\frac {1}{6}} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{8 \, b} - \frac {3 i \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{16 \, b^{\frac {3}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} + \frac {3 i \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{16 \, b^{\frac {3}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 67, normalized size = 0.66 \[ \sqrt {x}+\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.27, size = 95, normalized size = 0.93 \[ \frac {4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (3 i + 3\right ) \, \cos \left (2 \, a\right ) + \left (3 i - 3\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (\left (3 i - 3\right ) \, \cos \left (2 \, a\right ) - \left (3 i + 3\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 64 \, b^{3} \sqrt {x} + 48 \, b^{2} x^{\frac {1}{6}} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{64 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{\sqrt {x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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