Optimal. Leaf size=25 \[ -\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3207, 2564, 14} \[ -\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 3207
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{\sqrt {\sin ^3(x)}} \, dx &=\frac {\sin ^{\frac {3}{2}}(x) \int \frac {\cos ^3(x)}{\sin ^{\frac {3}{2}}(x)} \, dx}{\sqrt {\sin ^3(x)}}\\ &=\frac {\sin ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1-x^2}{x^{3/2}} \, dx,x,\sin (x)\right )}{\sqrt {\sin ^3(x)}}\\ &=\frac {\sin ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}-\sqrt {x}\right ) \, dx,x,\sin (x)\right )}{\sqrt {\sin ^3(x)}}\\ &=-\frac {2 \sin (x)}{\sqrt {\sin ^3(x)}}-\frac {2}{3} \sqrt {\sin ^3(x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.80 \[ \frac {\sin (x) (\cos (2 x)-7)}{3 \sqrt {\sin ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 28, normalized size = 1.12 \[ -\frac {2 \, {\left (\cos \relax (x)^{2} - 4\right )} \sqrt {-{\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}}{3 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 13, normalized size = 0.52 \[ -\frac {2}{3} \, \sin \relax (x)^{\frac {3}{2}} - \frac {2}{\sqrt {\sin \relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 14, normalized size = 0.56 \[ -\frac {2 \left (\sin ^{\frac {3}{2}}\relax (x )\right )}{3}-\frac {2}{\sqrt {\sin \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 19, normalized size = 0.76 \[ -\frac {2}{3} \, \sqrt {\sin \relax (x)^{3}} - \frac {2 \, \sin \relax (x)}{\sqrt {\sin \relax (x)^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\cos \relax (x)}^3}{\sqrt {{\sin \relax (x)}^3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.07, size = 36, normalized size = 1.44 \[ - \frac {8 \sin ^{3}{\relax (x )}}{3 \sqrt {\sin ^{3}{\relax (x )}}} - \frac {2 \sin {\relax (x )} \cos ^{2}{\relax (x )}}{\sqrt {\sin ^{3}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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