Optimal. Leaf size=68 \[ -\frac{\sin (x) \cos (x)}{a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}} \]
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Rubi [A] time = 0.0195453, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3767} \[ -\frac{\sin (x) \cos (x)}{a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (a \sin ^4(x)\right )^{3/2}} \, dx &=\frac{\sin ^2(x) \int \csc ^6(x) \, dx}{a \sqrt{a \sin ^4(x)}}\\ &=-\frac{\sin ^2(x) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (x)\right )}{a \sqrt{a \sin ^4(x)}}\\ &=-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{\cos (x) \sin (x)}{a \sqrt{a \sin ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0334892, size = 34, normalized size = 0.5 \[ -\frac{\sin ^5(x) \cos (x) \left (3 \csc ^4(x)+4 \csc ^2(x)+8\right )}{15 \left (a \sin ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 29, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( x \right ) \right ) ^{4}-20\, \left ( \cos \left ( x \right ) \right ) ^{2}+15 \right ) \sin \left ( x \right ) \cos \left ( x \right ) }{15} \left ( a \left ( \sin \left ( x \right ) \right ) ^{4} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44756, size = 31, normalized size = 0.46 \begin{align*} -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, a^{\frac{3}{2}} \tan \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62095, size = 196, normalized size = 2.88 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}{\left (8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )}}{15 \,{\left (a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}\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}{\left (a \sin ^{4}{\left (x \right )}\right )^{\frac{3}{2}}}\, 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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