Optimal. Leaf size=61 \[ -\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3204, 3207, 3770} \[ -\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sin ^2(x)\right )^{5/2}} \, dx &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}+\frac {3 \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx}{4 a}\\ &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}+\frac {3 \int \frac {1}{\sqrt {a \sin ^2(x)}} \, dx}{8 a^2}\\ &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}+\frac {(3 \sin (x)) \int \csc (x) \, dx}{8 a^2 \sqrt {a \sin ^2(x)}}\\ &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {3 \tanh ^{-1}(\cos (x)) \sin (x)}{8 a^2 \sqrt {a \sin ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 77, normalized size = 1.26 \[ -\frac {\csc (x) \sqrt {a \sin ^2(x)} \left (\csc ^4\left (\frac {x}{2}\right )+6 \csc ^2\left (\frac {x}{2}\right )-\sec ^4\left (\frac {x}{2}\right )-6 \sec ^2\left (\frac {x}{2}\right )+24 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )\right )}{64 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 78, normalized size = 1.28 \[ \frac {\sqrt {-a \cos \relax (x)^{2} + a} {\left (6 \, \cos \relax (x)^{3} + 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) - 10 \, \cos \relax (x)\right )}}{16 \, {\left (a^{3} \cos \relax (x)^{4} - 2 \, a^{3} \cos \relax (x)^{2} + a^{3}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 80, normalized size = 1.31 \[ \frac {\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{4} + 8 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \frac {12 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )} - \frac {18 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 8 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{4}}}{64 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.45, size = 89, normalized size = 1.46 \[ -\frac {\sqrt {a \left (\cos ^{2}\relax (x )\right )}\, \left (3 \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\relax (x )\right )}+2 a}{\sin \relax (x )}\right ) a \left (\sin ^{4}\relax (x )\right )+3 \sqrt {a \left (\cos ^{2}\relax (x )\right )}\, \left (\sin ^{2}\relax (x )\right ) \sqrt {a}+2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\relax (x )\right )}\right )}{8 a^{\frac {7}{2}} \sin \relax (x )^{3} \cos \relax (x ) \sqrt {a \left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.87, size = 931, normalized size = 15.26 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (a\,{\sin \relax (x)}^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sin ^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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