Integrand size = 13, antiderivative size = 37 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)} \]
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\frac {1}{2} i \left (4 i+3 \text {csch}(x)-3 \text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)} \text {csch}(x)+2 i \text {csch}^2(x)+\text {csch}^3(x)\right ) \tanh (x) \]
(I/2)*(4*I + 3*Csch[x] - 3*ArcTanh[Sqrt[Cosh[x]^2]]*Sqrt[Cosh[x]^2]*Csch[x ] + (2*I)*Csch[x]^2 + Csch[x]^3)*Tanh[x]
Time = 0.44 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.538, Rules used = {3042, 26, 26, 3247, 26, 3042, 26, 3227, 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.
\(\displaystyle \int \frac {\text {csch}^3(x)}{\sinh (x)+i} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{(i-i \sin (i x)) \sin (i x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int -\frac {i}{(1-\sin (i x)) \sin (i x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\int \frac {1}{(1-\sin (i x)) \sin (i x)^3}dx\) |
\(\Big \downarrow \) 3247 |
\(\displaystyle \int -i \text {csch}^3(x) (2 i \sinh (x)+3)dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \text {csch}^3(x) (2 i \sinh (x)+3)dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \int -\frac {i (2 \sin (i x)+3)}{\sin (i x)^3}dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\int \frac {2 \sin (i x)+3}{\sin (i x)^3}dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle -3 \int i \text {csch}^3(x)dx-2 \int -\text {csch}^2(x)dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 \int i \text {csch}^3(x)dx+2 \int \text {csch}^2(x)dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 i \int \text {csch}^3(x)dx+2 \int \text {csch}^2(x)dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int -\csc (i x)^2dx-3 i \int -i \csc (i x)^3dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \csc (i x)^2dx-3 i \int -i \csc (i x)^3dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -2 \int \csc (i x)^2dx-3 \int \csc (i x)^3dx-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -3 \int \csc (i x)^3dx-2 i \int 1d(-i \coth (x))-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -3 \int \csc (i x)^3dx-2 \coth (x)-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -3 \left (\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )-2 \coth (x)-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 \left (-\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )-2 \coth (x)-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -3 \left (-\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )-2 \coth (x)-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -3 \left (\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )-2 \coth (x)-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -3 \left (\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )-2 \coth (x)-\frac {i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\) |
-2*Coth[x] - 3*((I/2)*ArcTanh[Cosh[x]] - (I/2)*Coth[x]*Csch[x]) - (I*Coth[ x]*Csch[x])/(1 - I*Sinh[x])
3.1.46.3.1 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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[ c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
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]
Time = 3.50 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {2}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}\) | \(53\) |
risch | \(\frac {i \left (3 \,{\mathrm e}^{4 x}-5 \,{\mathrm e}^{2 x}+3 i {\mathrm e}^{3 x}+4-i {\mathrm e}^{x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left ({\mathrm e}^{x}+i\right )}+\frac {3 i \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {3 i \ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(62\) |
parallelrisch | \(\frac {\left (12 i \tanh \left (\frac {x}{2}\right )-12\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )-i \tanh \left (\frac {x}{2}\right )^{3}-3 i \coth \left (\frac {x}{2}\right )-\coth \left (\frac {x}{2}\right )^{2}-3 \tanh \left (\frac {x}{2}\right )^{2}-24}{8 \tanh \left (\frac {x}{2}\right )+8 i}\) | \(62\) |
-1/2*tanh(1/2*x)-1/8*I*tanh(1/2*x)^2-2/(tanh(1/2*x)+I)+1/8*I/tanh(1/2*x)^2 +3/2*I*ln(tanh(1/2*x))-1/2/tanh(1/2*x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.41 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3 \, {\left (i \, e^{\left (5 \, x\right )} - e^{\left (4 \, x\right )} - 2 i \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + 3 \, {\left (-i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} + 2 i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} - i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 6 i \, e^{\left (4 \, x\right )} + 6 \, e^{\left (3 \, x\right )} + 10 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} - 8 i}{2 \, {\left (e^{\left (5 \, x\right )} + i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + i\right )}} \]
-1/2*(3*(I*e^(5*x) - e^(4*x) - 2*I*e^(3*x) + 2*e^(2*x) + I*e^x - 1)*log(e^ x + 1) + 3*(-I*e^(5*x) + e^(4*x) + 2*I*e^(3*x) - 2*e^(2*x) - I*e^x + 1)*lo g(e^x - 1) - 6*I*e^(4*x) + 6*e^(3*x) + 10*I*e^(2*x) - 2*e^x - 8*I)/(e^(5*x ) + I*e^(4*x) - 2*e^(3*x) - 2*I*e^(2*x) + e^x + I)
\[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (27) = 54\).
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {e^{\left (-x\right )} + 5 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} - 4 i}{e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} - i \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} - i} - \frac {3}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
-(e^(-x) + 5*I*e^(-2*x) - 3*e^(-3*x) - 3*I*e^(-4*x) - 4*I)/(e^(-x) + 2*I*e ^(-2*x) - 2*e^(-3*x) - I*e^(-4*x) + e^(-5*x) - I) - 3/2*I*log(e^(-x) + 1) + 3/2*I*log(e^(-x) - 1)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\frac {i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + \frac {2 i}{e^{x} + i} - \frac {3}{2} i \, \log \left (e^{x} + 1\right ) + \frac {3}{2} i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
(I*e^(3*x) - 2*e^(2*x) + I*e^x + 2)/(e^(2*x) - 1)^2 + 2*I/(e^x + I) - 3/2* I*log(e^x + 1) + 3/2*I*log(abs(e^x - 1))
Time = 1.45 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}-3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {{\mathrm {e}}^x\,2{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {-2+{\mathrm {e}}^x\,1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]