Optimal. Leaf size=49 \[ -\frac {3}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right )+x^{3/2} \sin (x)+\frac {3}{2} \sqrt {x} \cos (x) \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3296, 3304, 3352} \[ -\frac {3}{2} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right )+x^{3/2} \sin (x)+\frac {3}{2} \sqrt {x} \cos (x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3352
Rubi steps
\begin {align*} \int x^{3/2} \cos (x) \, dx &=x^{3/2} \sin (x)-\frac {3}{2} \int \sqrt {x} \sin (x) \, dx\\ &=\frac {3}{2} \sqrt {x} \cos (x)+x^{3/2} \sin (x)-\frac {3}{4} \int \frac {\cos (x)}{\sqrt {x}} \, dx\\ &=\frac {3}{2} \sqrt {x} \cos (x)+x^{3/2} \sin (x)-\frac {3}{2} \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {3}{2} \sqrt {x} \cos (x)-\frac {3}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right )+x^{3/2} \sin (x)\\ \end {align*}
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Mathematica [C] time = 0.01, size = 55, normalized size = 1.12 \[ \frac {\sqrt {x} \Gamma \left (\frac {5}{2},-i x\right )}{2 \sqrt {-i x}}+\frac {\sqrt {x} \Gamma \left (\frac {5}{2},i x\right )}{2 \sqrt {i x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 35, normalized size = 0.71 \[ -\frac {3}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\pi }}\right ) + \frac {1}{2} \, {\left (2 \, x \sin \relax (x) + 3 \, \cos \relax (x)\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.49, size = 69, normalized size = 1.41 \[ \left (\frac {3}{16} i + \frac {3}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) - \left (\frac {3}{16} i - \frac {3}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) - \frac {1}{4} \, {\left (2 i \, x^{\frac {3}{2}} - 3 \, \sqrt {x}\right )} e^{\left (i \, x\right )} - \frac {1}{4} \, {\left (-2 i \, x^{\frac {3}{2}} - 3 \, \sqrt {x}\right )} e^{\left (-i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 0.69 \[ x^{\frac {3}{2}} \sin \relax (x )-\frac {3 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{4}+\frac {3 \sqrt {x}\, \cos \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.33, size = 74, normalized size = 1.51 \[ x^{\frac {3}{2}} \sin \relax (x) + \frac {1}{32} \, \sqrt {\pi } {\left (\left (3 i - 3\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) + \left (3 i + 3\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) - \left (3 i + 3\right ) \, \sqrt {2} \operatorname {erf}\left (\sqrt {-i} \sqrt {x}\right ) + \left (3 i - 3\right ) \, \sqrt {2} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} \sqrt {x}\right )\right )} + \frac {3}{2} \, \sqrt {x} \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{3/2}\,\cos \relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.78, size = 83, normalized size = 1.69 \[ \frac {5 x^{\frac {3}{2}} \sin {\relax (x )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {15 \sqrt {x} \cos {\relax (x )} \Gamma \left (\frac {5}{4}\right )}{8 \Gamma \left (\frac {9}{4}\right )} - \frac {15 \sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{16 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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