Optimal. Leaf size=70 \[ 2 i \sqrt {x} \text {Li}_2\left (-e^{2 i \sqrt {x}}\right )-\text {Li}_3\left (-e^{2 i \sqrt {x}}\right )+\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3747, 3719, 2190, 2531, 2282, 6589} \[ 2 i \sqrt {x} \text {PolyLog}\left (2,-e^{2 i \sqrt {x}}\right )-\text {PolyLog}\left (3,-e^{2 i \sqrt {x}}\right )+\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3719
Rule 3747
Rule 6589
Rubi steps
\begin {align*} \int \sqrt {x} \tan \left (\sqrt {x}\right ) \, dx &=2 \operatorname {Subst}\left (\int x^2 \tan (x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} i x^{3/2}-4 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right )+4 \operatorname {Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right )+2 i \sqrt {x} \text {Li}_2\left (-e^{2 i \sqrt {x}}\right )-2 i \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right )+2 i \sqrt {x} \text {Li}_2\left (-e^{2 i \sqrt {x}}\right )-\operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sqrt {x}}\right )\\ &=\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right )+2 i \sqrt {x} \text {Li}_2\left (-e^{2 i \sqrt {x}}\right )-\text {Li}_3\left (-e^{2 i \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 70, normalized size = 1.00 \[ 2 i \sqrt {x} \text {Li}_2\left (-e^{2 i \sqrt {x}}\right )-\text {Li}_3\left (-e^{2 i \sqrt {x}}\right )+\frac {2}{3} i x^{3/2}-2 x \log \left (1+e^{2 i \sqrt {x}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {x} \tan \left (\sqrt {x}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x} \tan \left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \sqrt {x}\, \tan \left (\sqrt {x}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 80, normalized size = 1.14 \[ -2 i \, x \arctan \left (\sin \left (2 \, \sqrt {x}\right ), \cos \left (2 \, \sqrt {x}\right ) + 1\right ) - x \log \left (\cos \left (2 \, \sqrt {x}\right )^{2} + \sin \left (2 \, \sqrt {x}\right )^{2} + 2 \, \cos \left (2 \, \sqrt {x}\right ) + 1\right ) + \frac {2}{3} i \, x^{\frac {3}{2}} + 2 i \, \sqrt {x} {\rm Li}_2\left (-e^{\left (2 i \, \sqrt {x}\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 i \, \sqrt {x}\right )}) \]
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 \[ \int \sqrt {x}\,\mathrm {tan}\left (\sqrt {x}\right ) \,d x \]
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 \[ \int \sqrt {x} \tan {\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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