Optimal. Leaf size=24 \[ -\tanh ^{-1}\left (\cos \left (\sqrt {x}\right )\right )-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4205, 3768, 3770} \[ -\tanh ^{-1}\left (\cos \left (\sqrt {x}\right )\right )-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rule 4205
Rubi steps
\begin {align*} \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \csc ^3(x) \, dx,x,\sqrt {x}\right )\\ &=-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )+\operatorname {Subst}\left (\int \csc (x) \, dx,x,\sqrt {x}\right )\\ &=-\tanh ^{-1}\left (\cos \left (\sqrt {x}\right )\right )-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [B] time = 0.04, size = 57, normalized size = 2.38 \[ -\frac {1}{4} \csc ^2\left (\frac {\sqrt {x}}{2}\right )+\frac {1}{4} \sec ^2\left (\frac {\sqrt {x}}{2}\right )+\log \left (\sin \left (\frac {\sqrt {x}}{2}\right )\right )-\log \left (\cos \left (\frac {\sqrt {x}}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 56, normalized size = 2.33 \[ -\frac {{\left (\cos \left (\sqrt {x}\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (\sqrt {x}\right ) + \frac {1}{2}\right ) - {\left (\cos \left (\sqrt {x}\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (\sqrt {x}\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (\sqrt {x}\right )}{2 \, {\left (\cos \left (\sqrt {x}\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 70, normalized size = 2.92 \[ -\frac {{\left (\frac {2 \, {\left (\cos \left (\sqrt {x}\right ) - 1\right )}}{\cos \left (\sqrt {x}\right ) + 1} - 1\right )} {\left (\cos \left (\sqrt {x}\right ) + 1\right )}}{4 \, {\left (\cos \left (\sqrt {x}\right ) - 1\right )}} - \frac {\cos \left (\sqrt {x}\right ) - 1}{4 \, {\left (\cos \left (\sqrt {x}\right ) + 1\right )}} + \frac {1}{2} \, \log \left (-\frac {\cos \left (\sqrt {x}\right ) - 1}{\cos \left (\sqrt {x}\right ) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 24, normalized size = 1.00 \[ -\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )+\ln \left (\csc \left (\sqrt {x}\right )-\cot \left (\sqrt {x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 34, normalized size = 1.42 \[ \frac {\cos \left (\sqrt {x}\right )}{\cos \left (\sqrt {x}\right )^{2} - 1} - \frac {1}{2} \, \log \left (\cos \left (\sqrt {x}\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cos \left (\sqrt {x}\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.33, size = 94, normalized size = 3.92 \[ -\ln \left (-\frac {{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{\sqrt {x}}-\frac {1{}\mathrm {i}}{\sqrt {x}}\right )+\ln \left (-\frac {{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{\sqrt {x}}+\frac {1{}\mathrm {i}}{\sqrt {x}}\right )+\frac {4\,{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}}{1+{\mathrm {e}}^{\sqrt {x}\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}}+\frac {2\,{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}}{{\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}-1} \]
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 \[ \int \frac {\csc ^{3}{\left (\sqrt {x} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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