Optimal. Leaf size=50 \[ -\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{3 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3769, 3771, 2641} \[ -\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{3 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx &=\frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{3/2}} \, dx}{\sqrt {a \csc ^3(x)}}\\ &=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}+\frac {(-\csc (x))^{3/2} \int \sqrt {-\csc (x)} \, dx}{3 \sqrt {a \csc ^3(x)}}\\ &=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}+\frac {\int \frac {1}{\sqrt {\sin (x)}} \, dx}{3 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{3 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 38, normalized size = 0.76 \[ \frac {-2 \cot (x)-\frac {2 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )}{\sin ^{\frac {3}{2}}(x)}}{3 \sqrt {a \csc ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \csc \relax (x)^{3}}}{a \csc \relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \csc \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.09, size = 125, normalized size = 2.50 \[ -\frac {\left (i \sin \relax (x ) \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-i-\sin \relax (x )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+2 \left (\cos ^{2}\relax (x )\right )-2 \cos \relax (x )\right ) \sqrt {8}}{6 \left (-1+\cos \relax (x )\right ) \sqrt {-\frac {2 a}{\sin \relax (x ) \left (-1+\cos ^{2}\relax (x )\right )}}\, \sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc \relax (x)^{3}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {a}{{\sin \relax (x)}^3}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \csc ^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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