Optimal. Leaf size=38 \[ x-\frac {2 i \cosh ^3(x)}{3 (1-i \sinh (x))^3}+\frac {2 i \cosh (x)}{1-i \sinh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4391, 2680, 8} \[ x-\frac {2 i \cosh ^3(x)}{3 (1-i \sinh (x))^3}+\frac {2 i \cosh (x)}{1-i \sinh (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2680
Rule 4391
Rubi steps
\begin {align*} \int \frac {1}{(\text {sech}(x)-i \tanh (x))^4} \, dx &=\int \frac {\cosh ^4(x)}{(1-i \sinh (x))^4} \, dx\\ &=-\frac {2 i \cosh ^3(x)}{3 (1-i \sinh (x))^3}-\int \frac {\cosh ^2(x)}{(1-i \sinh (x))^2} \, dx\\ &=-\frac {2 i \cosh ^3(x)}{3 (1-i \sinh (x))^3}+\frac {2 i \cosh (x)}{1-i \sinh (x)}+\int 1 \, dx\\ &=x-\frac {2 i \cosh ^3(x)}{3 (1-i \sinh (x))^3}+\frac {2 i \cosh (x)}{1-i \sinh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 74, normalized size = 1.95 \[ \frac {3 (3 x-8 i) \cosh \left (\frac {x}{2}\right )+(-3 x+16 i) \cosh \left (\frac {3 x}{2}\right )-6 i \sinh \left (\frac {x}{2}\right ) (2 x+x \cosh (x)-4 i)}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 52, normalized size = 1.37 \[ \frac {3 \, x e^{\left (3 \, x\right )} + {\left (9 i \, x + 24 i\right )} e^{\left (2 \, x\right )} - 3 \, {\left (3 \, x + 8\right )} e^{x} - 3 i \, x - 16 i}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 22, normalized size = 0.58 \[ x - \frac {-24 i \, e^{\left (2 \, x\right )} + 24 \, e^{x} + 16 i}{3 \, {\left (e^{x} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 41, normalized size = 1.08 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {8 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {16}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 40, normalized size = 1.05 \[ x - \frac {24 \, e^{\left (-x\right )} + 24 i \, e^{\left (-2 \, x\right )} - 16 i}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.61, size = 65, normalized size = 1.71 \[ x+\frac {\frac {{\mathrm {e}}^{2\,x}\,8{}\mathrm {i}}{3}-\frac {8}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}}+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________