Optimal. Leaf size=70 \[ -\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}+\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-a x \tan ^2(x) \sqrt{a \cot ^4(x)}-a \tan (x) \sqrt{a \cot ^4(x)} \]
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Rubi [A] time = 0.0271455, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 8} \[ -\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}+\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-a x \tan ^2(x) \sqrt{a \cot ^4(x)}-a \tan (x) \sqrt{a \cot ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \left (a \cot ^4(x)\right )^{3/2} \, dx &=\left (a \sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^6(x) \, dx\\ &=-\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}-\left (a \sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^4(x) \, dx\\ &=\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}+\left (a \sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^2(x) \, dx\\ &=\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}-a \sqrt{a \cot ^4(x)} \tan (x)-\left (a \sqrt{a \cot ^4(x)} \tan ^2(x)\right ) \int 1 \, dx\\ &=\frac{1}{3} a \cot (x) \sqrt{a \cot ^4(x)}-\frac{1}{5} a \cot ^3(x) \sqrt{a \cot ^4(x)}-a \sqrt{a \cot ^4(x)} \tan (x)-a x \sqrt{a \cot ^4(x)} \tan ^2(x)\\ \end{align*}
Mathematica [A] time = 0.135134, size = 39, normalized size = 0.56 \[ -\frac{1}{15} \tan ^6(x) \left (a \cot ^4(x)\right )^{3/2} \left (15 x+\cot (x) \left (3 \csc ^4(x)-11 \csc ^2(x)+23\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 40, normalized size = 0.6 \begin{align*}{\frac{1}{15\, \left ( \cot \left ( x \right ) \right ) ^{6}} \left ( a \left ( \cot \left ( x \right ) \right ) ^{4} \right ) ^{{\frac{3}{2}}} \left ( -3\, \left ( \cot \left ( x \right ) \right ) ^{5}+5\, \left ( \cot \left ( x \right ) \right ) ^{3}+{\frac{15\,\pi }{2}}-15\,{\rm arccot} \left (\cot \left ( x \right ) \right )-15\,\cot \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65532, size = 50, normalized size = 0.71 \begin{align*} -a^{\frac{3}{2}} x - \frac{15 \, a^{\frac{3}{2}} \tan \left (x\right )^{4} - 5 \, a^{\frac{3}{2}} \tan \left (x\right )^{2} + 3 \, a^{\frac{3}{2}}}{15 \, \tan \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17736, size = 289, normalized size = 4.13 \begin{align*} \frac{{\left (23 \, a \cos \left (2 \, x\right )^{3} - a \cos \left (2 \, x\right )^{2} - 11 \, a \cos \left (2 \, x\right ) + 15 \,{\left (a x \cos \left (2 \, x\right )^{2} - 2 \, a x \cos \left (2 \, x\right ) + a x\right )} \sin \left (2 \, x\right ) + 13 \, a\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 (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, 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 \left (a \cot ^{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 [A] time = 1.23163, size = 77, normalized size = 1.1 \begin{align*} \frac{1}{480} \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{5} - 35 \, \tan \left (\frac{1}{2} \, x\right )^{3} - 480 \, x - \frac{330 \, \tan \left (\frac{1}{2} \, x\right )^{4} - 35 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 3}{\tan \left (\frac{1}{2} \, x\right )^{5}} + 330 \, \tan \left (\frac{1}{2} \, x\right )\right )} a^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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