Optimal. Leaf size=16 \[ -\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}} \]
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Rubi [A] time = 0.0136734, antiderivative size = 16, 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, 3767, 8} \[ -\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sin ^4(x)}} \, dx &=\frac{\sin ^2(x) \int \csc ^2(x) \, dx}{\sqrt{a \sin ^4(x)}}\\ &=-\frac{\sin ^2(x) \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{\sqrt{a \sin ^4(x)}}\\ &=-\frac{\cos (x) \sin (x)}{\sqrt{a \sin ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0071261, size = 16, normalized size = 1. \[ -\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.159, size = 15, normalized size = 0.9 \begin{align*} -{\sin \left ( x \right ) \cos \left ( x \right ){\frac{1}{\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 = 1.4513, size = 12, normalized size = 0.75 \begin{align*} -\frac{1}{\sqrt{a} \tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54785, size = 95, normalized size = 5.94 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a} \cos \left (x\right )}{{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\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 \frac{1}{\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 [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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