Optimal. Leaf size=17 \[ -\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5} \]
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Rubi [A] time = 0.0260087, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2607, 14} \[ -\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \cot ^4(x) \csc ^4(x) \, dx &=\operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (x)\right )\\ &=-\frac{1}{5} \cot ^5(x)-\frac{\cot ^7(x)}{7}\\ \end{align*}
Mathematica [B] time = 0.0266781, size = 37, normalized size = 2.18 \[ -\frac{2 \cot (x)}{35}-\frac{1}{7} \cot (x) \csc ^6(x)+\frac{8}{35} \cot (x) \csc ^4(x)-\frac{1}{35} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 22, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{7\, \left ( \sin \left ( x \right ) \right ) ^{7}}}-{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{5}}{35\, \left ( \sin \left ( x \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.933675, size = 19, normalized size = 1.12 \begin{align*} -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95796, size = 112, normalized size = 6.59 \begin{align*} -\frac{2 \, \cos \left (x\right )^{7} - 7 \, \cos \left (x\right )^{5}}{35 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.066887, size = 41, normalized size = 2.41 \begin{align*} - \frac{2 \cos{\left (x \right )}}{35 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{35 \sin ^{3}{\left (x \right )}} + \frac{8 \cos{\left (x \right )}}{35 \sin ^{5}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{7 \sin ^{7}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07736, size = 19, normalized size = 1.12 \begin{align*} -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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