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 = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.50 \[ -\frac {\sin ^5(x) \cos (x) \left (3 \csc ^4(x)+4 \csc ^2(x)+8\right )}{15 \left (a \sin ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 74, normalized size = 1.09 \[ \frac {\sqrt {a \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} + a} {\left (8 \, \cos \relax (x)^{5} - 20 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)\right )}}{15 \, {\left (a^{2} \cos \relax (x)^{6} - 3 \, a^{2} \cos \relax (x)^{4} + 3 \, a^{2} \cos \relax (x)^{2} - a^{2}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 29, normalized size = 0.43 \[ -\frac {\left (8 \left (\cos ^{4}\relax (x )\right )-20 \left (\cos ^{2}\relax (x )\right )+15\right ) \sin \relax (x ) \cos \relax (x )}{15 \left (a \left (\sin ^{4}\relax (x )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 23, normalized size = 0.34 \[ -\frac {15 \, \tan \relax (x)^{4} + 10 \, \tan \relax (x)^{2} + 3}{15 \, a^{\frac {3}{2}} \tan \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.28, size = 44, normalized size = 0.65 \[ \frac {\frac {8{}\mathrm {i}}{15\,a^{3/2}}-\frac {4\,\left (2\,{\sin \left (2\,x\right )}^3-9\,\sin \left (2\,x\right )+3\,\sin \left (4\,x\right )+2{}\mathrm {i}\right )}{15\,a^{3/2}}}{{\left (\cos \left (2\,x\right )-1\right )}^3} \]
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 \[ \int \frac {1}{\left (a \sin ^{4}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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