Integrand size = 13, antiderivative size = 37 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3}{2} \text {arctanh}(\cosh (x))+2 i \coth (x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3903, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3}{2} \text {arctanh}(\cosh (x))+2 i \coth (x)+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}-\frac {3}{2} \coth (x) \text {csch}(x) \]
[In]
[Out]
Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 3903
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}-\int (2 i-3 \text {csch}(x)) \text {csch}^2(x) \, dx \\ & = \frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}-2 i \int \text {csch}^2(x) \, dx+3 \int \text {csch}^3(x) \, dx \\ & = -\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)}-\frac {3}{2} \int \text {csch}(x) \, dx-2 \text {Subst}(\int 1 \, dx,x,-i \coth (x)) \\ & = \frac {3}{2} \text {arctanh}(\cosh (x))+2 i \coth (x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(37)=74\).
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.43 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{8} \left (4 i \coth \left (\frac {x}{2}\right )-\text {csch}^2\left (\frac {x}{2}\right )+12 \log \left (\cosh \left (\frac {x}{2}\right )\right )-12 \log \left (\sinh \left (\frac {x}{2}\right )\right )-\text {sech}^2\left (\frac {x}{2}\right )+\frac {16 \sinh \left (\frac {x}{2}\right )}{-i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}+4 i \tanh \left (\frac {x}{2}\right )\right ) \]
[In]
[Out]
Time = 0.95 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}\) | \(53\) |
risch | \(-\frac {-5 \,{\mathrm e}^{2 x}-3 i {\mathrm e}^{3 x}+3 \,{\mathrm e}^{4 x}+4+i {\mathrm e}^{x}}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left ({\mathrm e}^{x}-i\right )}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{2}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(59\) |
parallelrisch | \(\frac {\left (-12 i+12 \tanh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )-i \coth \left (\frac {x}{2}\right )^{2}-3 i \tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right )^{3}-24 i-3 \coth \left (\frac {x}{2}\right )}{-8 \tanh \left (\frac {x}{2}\right )+8 i}\) | \(63\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.24 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 3 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 6 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} - 8}{2 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + e^{x} - i\right )}} \]
[In]
[Out]
\[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (29) = 58\).
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.08 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=-\frac {-i \, e^{\left (-x\right )} - 5 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 4}{e^{\left (-x\right )} - 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + i \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + i} + \frac {3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.35 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=-\frac {e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + 2 i}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} - \frac {2 i}{i \, e^{x} + 1} + \frac {3}{2} \, \log \left (e^{x} + 1\right ) - \frac {3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
[In]
[Out]
Time = 2.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3\,\ln \left (3\,{\mathrm {e}}^x+3\right )}{2}-\frac {3\,\ln \left (3\,{\mathrm {e}}^x-3\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {2}{{\mathrm {e}}^x-\mathrm {i}}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]
[In]
[Out]