Integrand size = 13, antiderivative size = 52 \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=-i x+\frac {1}{15} (15 i-8 \text {csch}(x)) \tanh (x)+\frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x) \]
[Out]
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3973, 3967, 8} \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=-i x+\frac {1}{5} \tanh ^5(x) (-\text {csch}(x)+i)+\frac {1}{15} \tanh ^3(x) (-4 \text {csch}(x)+5 i)+\frac {1}{15} \tanh (x) (-8 \text {csch}(x)+15 i) \]
[In]
[Out]
Rule 8
Rule 3967
Rule 3973
Rubi steps \begin{align*} \text {integral}& = \int (-i+\text {csch}(x)) \tanh ^6(x) \, dx \\ & = \frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)-\frac {1}{5} \int (5 i-4 \text {csch}(x)) \tanh ^4(x) \, dx \\ & = \frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)+\frac {1}{15} \int (-15 i+8 \text {csch}(x)) \tanh ^2(x) \, dx \\ & = \frac {1}{15} (15 i-8 \text {csch}(x)) \tanh (x)+\frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x)-\frac {1}{15} \int 15 i \, dx \\ & = -i x+\frac {1}{15} (15 i-8 \text {csch}(x)) \tanh (x)+\frac {1}{15} (5 i-4 \text {csch}(x)) \tanh ^3(x)+\frac {1}{5} (i-\text {csch}(x)) \tanh ^5(x) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(52)=104\).
Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.42 \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {-200+6 (89-120 i x) \cosh (x)-128 \cosh (2 x)+178 \cosh (3 x)-240 i x \cosh (3 x)-184 \cosh (4 x)+64 i \sinh (x)+178 i \sinh (2 x)+240 x \sinh (2 x)+128 i \sinh (3 x)+89 i \sinh (4 x)+120 x \sinh (4 x)}{960 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^5} \]
[In]
[Out]
Time = 2.42 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.31
method | result | size |
risch | \(-i x -\frac {2 \left (-31 i {\mathrm e}^{2 x}+73 \,{\mathrm e}^{3 x}+31 \,{\mathrm e}^{x}-25 i {\mathrm e}^{4 x}+65 \,{\mathrm e}^{5 x}-23 i+15 i {\mathrm e}^{6 x}+15 \,{\mathrm e}^{7 x}\right )}{15 \left ({\mathrm e}^{x}+i\right )^{3} \left ({\mathrm e}^{x}-i\right )^{5}}\) | \(68\) |
default | \(i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {11 i}{8 \left (\tanh \left (\frac {x}{2}\right )-i\right )}+\frac {2 i}{5 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{5}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{4}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {5 i}{8 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {i}{6 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(99\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (36) = 72\).
Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.38 \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {-15 i \, x e^{\left (8 \, x\right )} - 30 \, {\left (x + 1\right )} e^{\left (7 \, x\right )} - 30 \, {\left (i \, x + i\right )} e^{\left (6 \, x\right )} - 10 \, {\left (9 \, x + 13\right )} e^{\left (5 \, x\right )} - 2 \, {\left (45 \, x + 73\right )} e^{\left (3 \, x\right )} - 2 \, {\left (-15 i \, x - 31 i\right )} e^{\left (2 \, x\right )} - 2 \, {\left (15 \, x + 31\right )} e^{x} + 15 i \, x + 50 i \, e^{\left (4 \, x\right )} + 46 i}{15 \, {\left (e^{\left (8 \, x\right )} - 2 i \, e^{\left (7 \, x\right )} + 2 \, e^{\left (6 \, x\right )} - 6 i \, e^{\left (5 \, x\right )} - 6 i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1\right )}} \]
[In]
[Out]
\[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\tanh ^{4}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (36) = 72\).
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.85 \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=-i \, x - \frac {2 \, {\left (31 \, e^{\left (-x\right )} + 31 i \, e^{\left (-2 \, x\right )} + 73 \, e^{\left (-3 \, x\right )} + 25 i \, e^{\left (-4 \, x\right )} + 65 \, e^{\left (-5 \, x\right )} - 15 i \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-7 \, x\right )} + 23 i\right )}}{30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} + 90 i \, e^{\left (-3 \, x\right )} + 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} + 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.12 \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=-i \, x - \frac {21 i \, e^{\left (2 \, x\right )} - 36 \, e^{x} - 19 i}{24 \, {\left (i \, e^{x} - 1\right )}^{3}} - \frac {115 \, e^{\left (4 \, x\right )} - 380 i \, e^{\left (3 \, x\right )} - 530 \, e^{\left (2 \, x\right )} + 340 i \, e^{x} + 91}{40 \, {\left (e^{x} - i\right )}^{5}} \]
[In]
[Out]
Time = 2.89 (sec) , antiderivative size = 237, normalized size of antiderivative = 4.56 \[ \int \frac {\tanh ^4(x)}{i+\text {csch}(x)} \, dx=-x\,1{}\mathrm {i}-\frac {1{}\mathrm {i}}{4\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {\frac {23\,{\mathrm {e}}^x}{40}-\frac {3}{8}{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {23}{40\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {7}{8\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}+\frac {\frac {{\mathrm {e}}^{2\,x}\,9{}\mathrm {i}}{8}-\frac {23\,{\mathrm {e}}^{3\,x}}{40}+\frac {9\,{\mathrm {e}}^x}{8}-\frac {3}{8}{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {\frac {3}{8}-\frac {23\,{\mathrm {e}}^{2\,x}}{40}+\frac {{\mathrm {e}}^x\,3{}\mathrm {i}}{4}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {1}{6\,\left ({\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}\right )}-\frac {\frac {23\,{\mathrm {e}}^{4\,x}}{40}-\frac {9\,{\mathrm {e}}^{2\,x}}{4}+\frac {23}{40}-\frac {{\mathrm {e}}^{3\,x}\,3{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^x\,3{}\mathrm {i}}{2}}{{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}-10\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,5{}\mathrm {i}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x-\mathrm {i}} \]
[In]
[Out]