Optimal. Leaf size=19 \[ -\frac{\cot ^2(x)}{2}+\frac{1}{3} i \cot ^3(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.038832, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3516, 848, 43} \[ -\frac{\cot ^2(x)}{2}+\frac{1}{3} i \cot ^3(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3516
Rule 848
Rule 43
Rubi steps
\begin{align*} \int \frac{\csc ^4(x)}{i+\tan (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 (i+x)} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{-i+x}{x^4} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{i}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac{1}{2} \cot ^2(x)+\frac{1}{3} i \cot ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0175136, size = 29, normalized size = 1.53 \[ -\frac{1}{3} i \cot (x)-\frac{\csc ^2(x)}{2}+\frac{1}{3} i \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.038, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\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 = 1.36622, size = 16, normalized size = 0.84 \begin{align*} -\frac{i \,{\left (-3 i \, \tan \left (x\right ) - 2\right )}}{6 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.99596, size = 109, normalized size = 5.74 \begin{align*} \frac{2 \,{\left (6 \, e^{\left (4 i \, x\right )} - 3 \, e^{\left (2 i \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (6 i \, x\right )} - 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.41731, size = 16, normalized size = 0.84 \begin{align*} -\frac{3 \, \tan \left (x\right ) - 2 i}{6 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]