Optimal. Leaf size=27 \[ -i x-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\frac {1}{2} \coth (x) (-\text {csch}(x)+2 i) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {3888, 3881, 3770} \[ -i x-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\frac {1}{2} \coth (x) (-\text {csch}(x)+2 i) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rule 3881
Rule 3888
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{i+\text {csch}(x)} \, dx &=\int \coth ^2(x) (-i+\text {csch}(x)) \, dx\\ &=\frac {1}{2} \coth (x) (2 i-\text {csch}(x))+\frac {1}{2} \int (-2 i+\text {csch}(x)) \, dx\\ &=-i x+\frac {1}{2} \coth (x) (2 i-\text {csch}(x))+\frac {1}{2} \int \text {csch}(x) \, dx\\ &=-i x-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\frac {1}{2} \coth (x) (2 i-\text {csch}(x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 65, normalized size = 2.41 \[ -i x+\frac {1}{2} i \tanh \left (\frac {x}{2}\right )+\frac {1}{2} i \coth \left (\frac {x}{2}\right )-\frac {1}{8} \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{8} \text {sech}^2\left (\frac {x}{2}\right )+\frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.06, size = 85, normalized size = 3.15 \[ \frac {-2 i \, x e^{\left (4 \, x\right )} + {\left (4 i \, x + 4 i\right )} e^{\left (2 \, x\right )} - {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + 1\right ) + {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i \, x - 2 \, e^{\left (3 \, x\right )} - 2 \, e^{x} - 4 i}{2 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.13, size = 48, normalized size = 1.78 \[ \frac {e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + 2 i}{{\left (i \, e^{\left (2 \, x\right )} - i\right )}^{2}} - i \, \log \left (-i \, e^{x}\right ) - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.18, size = 61, normalized size = 2.26 \[ \frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 55, normalized size = 2.04 \[ -i \, x + \frac {e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 2 i}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.64, size = 56, normalized size = 2.07 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2}-\frac {{\mathrm {e}}^x-2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-x\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{4}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________