Optimal. Leaf size=77 \[ -\frac{x \cot ^2(x)}{a \sqrt{a \cot ^4(x)}}+\frac{\cot (x)}{a \sqrt{a \cot ^4(x)}}+\frac{\tan ^3(x)}{5 a \sqrt{a \cot ^4(x)}}-\frac{\tan (x)}{3 a \sqrt{a \cot ^4(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0257301, antiderivative size = 77, 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} \[ -\frac{x \cot ^2(x)}{a \sqrt{a \cot ^4(x)}}+\frac{\cot (x)}{a \sqrt{a \cot ^4(x)}}+\frac{\tan ^3(x)}{5 a \sqrt{a \cot ^4(x)}}-\frac{\tan (x)}{3 a \sqrt{a \cot ^4(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3658
Rule 3473
Rule 8
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*}
Mathematica [A] time = 0.10534, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 42, normalized size = 0.6 \begin{align*}{\frac{\cot \left ( x \right ) }{15} \left ( 15\, \left ( \pi /2-{\rm arccot} \left (\cot \left ( x \right ) \right ) \right ) \left ( \cot \left ( x \right ) \right ) ^{5}+15\, \left ( \cot \left ( x \right ) \right ) ^{4}-5\, \left ( \cot \left ( x \right ) \right ) ^{2}+3 \right ) \left ( a \left ( \cot \left ( x \right ) \right ) ^{4} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.66123, size = 39, normalized size = 0.51 \begin{align*} \frac{3 \, \tan \left (x\right )^{5} - 5 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a^{\frac{3}{2}}} - \frac{x}{a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.62312, size = 367, normalized size = 4.77 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cot ^{4}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]