Integrand size = 13, antiderivative size = 37 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3}{2} \text {arctanh}(\cosh (x))+2 i \coth (x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\text {csch}(x)} \]
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 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{8} \left (4 i \coth \left (\frac {x}{2}\right )-\text {csch}^2\left (\frac {x}{2}\right )+12 \log \left (\cosh \left (\frac {x}{2}\right )\right )-12 \log \left (\sinh \left (\frac {x}{2}\right )\right )-\text {sech}^2\left (\frac {x}{2}\right )+\frac {16 \sinh \left (\frac {x}{2}\right )}{-i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}+4 i \tanh \left (\frac {x}{2}\right )\right ) \]
((4*I)*Coth[x/2] - Csch[x/2]^2 + 12*Log[Cosh[x/2]] - 12*Log[Sinh[x/2]] - S ech[x/2]^2 + (16*Sinh[x/2])/((-I)*Cosh[x/2] + Sinh[x/2]) + (4*I)*Tanh[x/2] )/8
Time = 0.45 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.462, Rules used = {3042, 4305, 25, 3042, 25, 26, 4274, 25, 26, 3042, 25, 26, 4254, 24, 4255, 26, 3042, 26, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\text {csch}^4(x)}{\text {csch}(x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc (i x)^4}{i \csc (i x)+i}dx\) |
\(\Big \downarrow \) 4305 |
\(\displaystyle \int -\left ((2 i-3 \text {csch}(x)) \text {csch}^2(x)\right )dx+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}-\int (2 i-3 \text {csch}(x)) \text {csch}^2(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}-\int -\left ((2 i-3 i \csc (i x)) \csc (i x)^2\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int i (2-3 \csc (i x)) \csc (i x)^2dx+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int (2-3 \csc (i x)) \csc (i x)^2dx+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle i \left (2 \int -\text {csch}^2(x)dx-3 \int i \text {csch}^3(x)dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-2 \int \text {csch}^2(x)dx-3 \int i \text {csch}^3(x)dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (-2 \int \text {csch}^2(x)dx-3 i \int \text {csch}^3(x)dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (-2 \int -\csc (i x)^2dx-3 i \int -i \csc (i x)^3dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (2 \int \csc (i x)^2dx-3 i \int -i \csc (i x)^3dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (2 \int \csc (i x)^2dx-3 \int \csc (i x)^3dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle i \left (2 i \int 1d(-i \coth (x))-3 \int \csc (i x)^3dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle i \left (2 \coth (x)-3 \int \csc (i x)^3dx\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle i \left (2 \coth (x)-3 \left (\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (2 \coth (x)-3 \left (-\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (2 \coth (x)-3 \left (-\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (2 \coth (x)-3 \left (\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle i \left (2 \coth (x)-3 \left (\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )\right )+\frac {\coth (x) \text {csch}^2(x)}{\text {csch}(x)+i}\) |
(Coth[x]*Csch[x]^2)/(I + Csch[x]) + I*(2*Coth[x] - 3*((I/2)*ArcTanh[Cosh[x ]] - (I/2)*Coth[x]*Csch[x]))
3.1.69.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[d^2*Cot[e + f*x]*((d*Csc[e + f*x])^(n - 2)/(f*(a + b*Csc[e + f*x]))), x] - Simp[d^2/(a*b) Int[(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) - a*(n - 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ [a^2 - b^2, 0] && GtQ[n, 1]
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 | \(\frac {\left (-12 i+12 \tanh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )-i \coth \left (\frac {x}{2}\right )^{2}-3 i \tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right )^{3}-24 i-3 \coth \left (\frac {x}{2}\right )}{-8 \tanh \left (\frac {x}{2}\right )+8 i}\) | \(63\) |
1/2*I*tanh(1/2*x)+1/8*tanh(1/2*x)^2-1/8/tanh(1/2*x)^2+1/2*I/tanh(1/2*x)-3/ 2*ln(tanh(1/2*x))+2*I/(tanh(1/2*x)-I)
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 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 3 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 6 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} - 8}{2 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} + e^{x} - i\right )}} \]
1/2*(3*(e^(5*x) - I*e^(4*x) - 2*e^(3*x) + 2*I*e^(2*x) + e^x - I)*log(e^x + 1) - 3*(e^(5*x) - I*e^(4*x) - 2*e^(3*x) + 2*I*e^(2*x) + e^x - I)*log(e^x - 1) - 6*e^(4*x) + 6*I*e^(3*x) + 10*e^(2*x) - 2*I*e^x - 8)/(e^(5*x) - I*e^ (4*x) - 2*e^(3*x) + 2*I*e^(2*x) + e^x - I)
\[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]
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 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=-\frac {-i \, e^{\left (-x\right )} - 5 \, e^{\left (-2 \, x\right )} + 3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 4}{e^{\left (-x\right )} - 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + i \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} + i} + \frac {3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
-(-I*e^(-x) - 5*e^(-2*x) + 3*I*e^(-3*x) + 3*e^(-4*x) + 4)/(e^(-x) - 2*I*e^ (-2*x) - 2*e^(-3*x) + I*e^(-4*x) + e^(-5*x) + I) + 3/2*log(e^(-x) + 1) - 3 /2*log(e^(-x) - 1)
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.35 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=-\frac {e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + 2 i}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} - \frac {2 i}{i \, e^{x} + 1} + \frac {3}{2} \, \log \left (e^{x} + 1\right ) - \frac {3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
-(e^(3*x) - 2*I*e^(2*x) + e^x + 2*I)/(e^(2*x) - 1)^2 - 2*I/(I*e^x + 1) + 3 /2*log(e^x + 1) - 3/2*log(abs(e^x - 1))
Time = 2.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70 \[ \int \frac {\text {csch}^4(x)}{i+\text {csch}(x)} \, dx=\frac {3\,\ln \left (3\,{\mathrm {e}}^x+3\right )}{2}-\frac {3\,\ln \left (3\,{\mathrm {e}}^x-3\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {2}{{\mathrm {e}}^x-\mathrm {i}}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]