Optimal. Leaf size=33 \[ -\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0391877, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3487, 43} \[ -\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\csc ^6(x)}{i+\cot (x)} \, dx &=-\operatorname{Subst}\left (\int (i-x)^2 (i+x) \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-i+x-i x^2+x^3\right ) \, dx,x,\cot (x)\right )\\ &=i \cot (x)-\frac{\cot ^2(x)}{2}+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^4(x)}{4}\\ \end{align*}
Mathematica [A] time = 0.0523081, size = 23, normalized size = 0.7 \[ -\frac{\csc ^4(x)}{4}+\frac{1}{3} i \cot (x) \left (\csc ^2(x)+2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.052, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{4\, \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{i}{\tan \left ( x \right ) }}+{\frac{{\frac{i}{3}}}{ \left ( \tan \left ( x \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.20775, size = 32, normalized size = 0.97 \begin{align*} \frac{12 i \, \tan \left (x\right )^{3} - 6 \, \tan \left (x\right )^{2} + 4 i \, \tan \left (x\right ) - 3}{12 \, \tan \left (x\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.63184, size = 228, normalized size = 6.91 \begin{align*} -\frac{4 \,{\left (2 \,{\left (10 \, e^{\left (4 i \, x\right )} - 5 \, e^{\left (2 i \, x\right )} + 1\right )} e^{\left (2 i \, x\right )} - 15 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{15 \,{\left (e^{\left (10 i \, x\right )} - 5 \, e^{\left (8 i \, x\right )} + 10 \, e^{\left (6 i \, x\right )} - 10 \, e^{\left (4 i \, x\right )} + 5 \, e^{\left (2 i \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16601, size = 32, normalized size = 0.97 \begin{align*} -\frac{-12 i \, \tan \left (x\right )^{3} + 6 \, \tan \left (x\right )^{2} - 4 i \, \tan \left (x\right ) + 3}{12 \, \tan \left (x\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]