Optimal. Leaf size=36 \[ \frac{x \csc ^2(x)}{2 \sqrt{a \csc ^4(x)}}-\frac{\cot (x)}{2 \sqrt{a \csc ^4(x)}} \]
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Rubi [A] time = 0.01555, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 2635, 8} \[ \frac{x \csc ^2(x)}{2 \sqrt{a \csc ^4(x)}}-\frac{\cot (x)}{2 \sqrt{a \csc ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \csc ^4(x)}} \, dx &=\frac{\csc ^2(x) \int \sin ^2(x) \, dx}{\sqrt{a \csc ^4(x)}}\\ &=-\frac{\cot (x)}{2 \sqrt{a \csc ^4(x)}}+\frac{\csc ^2(x) \int 1 \, dx}{2 \sqrt{a \csc ^4(x)}}\\ &=-\frac{\cot (x)}{2 \sqrt{a \csc ^4(x)}}+\frac{x \csc ^2(x)}{2 \sqrt{a \csc ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0252097, size = 25, normalized size = 0.69 \[ \frac{x \csc ^2(x)-\cot (x)}{2 \sqrt{a \csc ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.163, size = 24, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( x \right ) \sin \left ( x \right ) -x}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a}{ \left ( \sin \left ( x \right ) \right ) ^{4}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49717, size = 34, normalized size = 0.94 \begin{align*} \frac{x}{2 \, \sqrt{a}} - \frac{\tan \left (x\right )}{2 \,{\left (\sqrt{a} \tan \left (x\right )^{2} + \sqrt{a}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.488808, size = 122, normalized size = 3.39 \begin{align*} -\frac{{\left (x \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) - x\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \csc ^{4}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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