Optimal. Leaf size=36 \[ \frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)} \]
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Rubi [A] time = 0.0131223, 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 = {3207, 2635, 8} \[ \frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sqrt{a \sin ^4(x)} \, dx &=\left (\csc ^2(x) \sqrt{a \sin ^4(x)}\right ) \int \sin ^2(x) \, dx\\ &=-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}+\frac{1}{2} \left (\csc ^2(x) \sqrt{a \sin ^4(x)}\right ) \int 1 \, dx\\ &=-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}+\frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}\\ \end{align*}
Mathematica [A] time = 0.0164401, size = 25, normalized size = 0.69 \[ \frac{1}{2} \csc (x) \sqrt{a \sin ^4(x)} (x \csc (x)-\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 27, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{16} \left ( \sin \left ( x \right ) \cos \left ( x \right ) -x \right ) }{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{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 = 1.44084, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{2} \, \sqrt{a} x - \frac{\sqrt{a} \tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6705, size = 103, normalized size = 2.86 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}{\left (\cos \left (x\right ) \sin \left (x\right ) - x\right )}}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \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 \sqrt{a \sin ^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09986, size = 20, normalized size = 0.56 \begin{align*} \frac{1}{4} \, \sqrt{a}{\left (2 \, x - \sin \left (2 \, x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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