Optimal. Leaf size=37 \[ -2 \coth (x)-\frac {3}{2} i \tanh ^{-1}(\cosh (x))+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{\sinh (x)+i} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2768, 2748, 3768, 3770, 3767, 8} \[ -2 \coth (x)-\frac {3}{2} i \tanh ^{-1}(\cosh (x))+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2748
Rule 2768
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx &=\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}+\int \text {csch}^3(x) (-3 i+2 \sinh (x)) \, dx\\ &=\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}-3 i \int \text {csch}^3(x) \, dx+2 \int \text {csch}^2(x) \, dx\\ &=\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}+\frac {3}{2} i \int \text {csch}(x) \, dx-2 i \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-\frac {3}{2} i \tanh ^{-1}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 49, normalized size = 1.32 \[ \frac {1}{2} i \tanh (x) \left (\text {csch}^3(x)+2 i \text {csch}^2(x)+3 \text {csch}(x)-3 \sqrt {\cosh ^2(x)} \text {csch}(x) \tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right )+4 i\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.73, size = 129, normalized size = 3.49 \[ \frac {{\left (-3 i \, e^{\left (5 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 3 i \, e^{x} + 3\right )} \log \left (e^{x} + 1\right ) + {\left (3 i \, e^{\left (5 \, x\right )} - 3 \, e^{\left (4 \, x\right )} - 6 i \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 3\right )} \log \left (e^{x} - 1\right ) + 6 i \, e^{\left (4 \, x\right )} - 6 \, e^{\left (3 \, x\right )} - 10 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 8 i}{2 \, e^{\left (5 \, x\right )} + 2 i \, e^{\left (4 \, x\right )} - 4 \, e^{\left (3 \, x\right )} - 4 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 2 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 51, normalized size = 1.38 \[ \frac {i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + \frac {2 i}{e^{x} + i} - \frac {3}{2} i \, \log \left (e^{x} + 1\right ) + \frac {3}{2} i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 53, normalized size = 1.43 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {i \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {2}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.32, size = 79, normalized size = 2.14 \[ -\frac {8 \, {\left (e^{\left (-x\right )} + 5 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} - 4 i\right )}}{8 \, e^{\left (-x\right )} + 16 i \, e^{\left (-2 \, x\right )} - 16 \, e^{\left (-3 \, x\right )} - 8 i \, e^{\left (-4 \, x\right )} + 8 \, e^{\left (-5 \, x\right )} - 8 i} - \frac {3}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.66, size = 70, normalized size = 1.89 \[ -\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}-3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {{\mathrm {e}}^x\,2{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {-2+{\mathrm {e}}^x\,1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{3}{\relax (x )}}{\sinh {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________