Optimal. Leaf size=77 \[ \frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3658, 3473, 8} \[ -\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}+\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx &=\frac {\cot ^2(x) \int \tan ^6(x) \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\cot ^2(x) \int \tan ^4(x) \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}+\frac {\cot ^2(x) \int \tan ^2(x) \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\cot ^2(x) \int 1 \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 42, normalized size = 0.55 \[ \frac {-15 x \cot ^2(x)+23 \cot (x)+\csc (x) \sec (x) \left (3 \sec ^2(x)-11\right )}{15 a \sqrt {a \cot ^4(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.37, size = 142, normalized size = 1.84 \[ \frac {{\left (15 \, x \cos \left (2 \, x\right )^{4} + 30 \, x \cos \left (2 \, x\right )^{3} - 30 \, x \cos \left (2 \, x\right ) - {\left (23 \, \cos \left (2 \, x\right )^{3} + \cos \left (2 \, x\right )^{2} - 11 \, \cos \left (2 \, x\right ) - 13\right )} \sin \left (2 \, x\right ) - 15 \, x\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{15 \, {\left (a^{2} \cos \left (2 \, x\right )^{4} + 4 \, a^{2} \cos \left (2 \, x\right )^{3} + 6 \, a^{2} \cos \left (2 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 43, normalized size = 0.56 \[ -\frac {\frac {15 \, x}{\sqrt {a}} - \frac {3 \, a^{2} \tan \relax (x)^{5} - 5 \, a^{2} \tan \relax (x)^{3} + 15 \, a^{2} \tan \relax (x)}{a^{\frac {5}{2}}}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 42, normalized size = 0.55 \[ \frac {\cot \relax (x ) \left (15 \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \relax (x )\right )\right ) \left (\cot ^{5}\relax (x )\right )+15 \left (\cot ^{4}\relax (x )\right )-5 \left (\cot ^{2}\relax (x )\right )+3\right )}{15 \left (a \left (\cot ^{4}\relax (x )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 29, normalized size = 0.38 \[ \frac {3 \, \tan \relax (x)^{5} - 5 \, \tan \relax (x)^{3} + 15 \, \tan \relax (x)}{15 \, a^{\frac {3}{2}}} - \frac {x}{a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,{\mathrm {cot}\relax (x)}^4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \cot ^{4}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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