Integrand size = 13, antiderivative size = 58 \[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {7}{2} \text {arctanh}(\cosh (x))+\frac {16}{3} i \coth (x)+\frac {7}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{3 (i+\sinh (x))^2}-\frac {8 i \coth (x) \text {csch}(x)}{3 (i+\sinh (x))} \] Output:
-7/2*arctanh(cosh(x))+16/3*I*coth(x)+7/2*coth(x)*csch(x)+1/3*coth(x)*csch( x)/(I+sinh(x))^2-8/3*I*coth(x)*csch(x)/(I+sinh(x))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(58)=116\).
Time = 1.02 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.41 \[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{24} \left (24 i \coth \left (\frac {x}{2}\right )+3 \text {csch}^2\left (\frac {x}{2}\right )-84 \log \left (\cosh \left (\frac {x}{2}\right )\right )+84 \log \left (\sinh \left (\frac {x}{2}\right )\right )+3 \text {sech}^2\left (\frac {x}{2}\right )+\frac {8}{\left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^2}+\frac {160 i \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )}+\frac {16 \sinh \left (\frac {x}{2}\right )}{\left (i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )^3}+24 i \tanh \left (\frac {x}{2}\right )\right ) \] Input:
Integrate[Csch[x]^3/(I + Sinh[x])^2,x]
Output:
((24*I)*Coth[x/2] + 3*Csch[x/2]^2 - 84*Log[Cosh[x/2]] + 84*Log[Sinh[x/2]] + 3*Sech[x/2]^2 + 8/(Cosh[x/2] - I*Sinh[x/2])^2 + ((160*I)*Sinh[x/2])/(Cos h[x/2] - I*Sinh[x/2]) + (16*Sinh[x/2])/(I*Cosh[x/2] + Sinh[x/2])^3 + (24*I )*Tanh[x/2])/24
Time = 0.67 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.846, Rules used = {3042, 26, 25, 3245, 26, 3042, 26, 3457, 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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{(\sinh (x)+i)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{(i-i \sin (i x))^2 \sin (i x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int -\frac {1}{(1-\sin (i x))^2 \sin (i x)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \int \frac {1}{(1-\sin (i x))^2 \sin (i x)^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle i \left (\frac {1}{3} \int \frac {i \text {csch}^3(x) (3 i \sinh (x)+5)}{1-i \sinh (x)}dx+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} i \int \frac {\text {csch}^3(x) (3 i \sinh (x)+5)}{1-i \sinh (x)}dx+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} i \int -\frac {i (3 \sin (i x)+5)}{(1-\sin (i x)) \sin (i x)^3}dx+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \int \frac {3 \sin (i x)+5}{(1-\sin (i x)) \sin (i x)^3}dx+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle i \left (\frac {1}{3} \left (\int i \text {csch}^3(x) (16 i \sinh (x)+21)dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \left (i \int \text {csch}^3(x) (16 i \sinh (x)+21)dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} \left (i \int -\frac {i (16 \sin (i x)+21)}{\sin (i x)^3}dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \left (\int \frac {16 \sin (i x)+21}{\sin (i x)^3}dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \int i \text {csch}^3(x)dx+16 \int -\text {csch}^2(x)dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \int i \text {csch}^3(x)dx-16 \int \text {csch}^2(x)dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 i \int \text {csch}^3(x)dx-16 \int \text {csch}^2(x)dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} \left (-16 \int -\csc (i x)^2dx+21 i \int -i \csc (i x)^3dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {1}{3} \left (16 \int \csc (i x)^2dx+21 i \int -i \csc (i x)^3dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \left (16 \int \csc (i x)^2dx+21 \int \csc (i x)^3dx+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \int \csc (i x)^3dx+16 i \int 1d(-i \coth (x))+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \int \csc (i x)^3dx+16 \coth (x)+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \left (\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+16 \coth (x)+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \left (-\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+16 \coth (x)+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \left (-\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+16 \coth (x)+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \left (\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+16 \coth (x)+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle i \left (\frac {1}{3} \left (21 \left (\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+16 \coth (x)+\frac {8 i \coth (x) \text {csch}(x)}{1-i \sinh (x)}\right )+\frac {i \coth (x) \text {csch}(x)}{3 (1-i \sinh (x))^2}\right )\) |
Input:
Int[Csch[x]^3/(I + Sinh[x])^2,x]
Output:
I*((16*Coth[x] + 21*((I/2)*ArcTanh[Cosh[x]] - (I/2)*Coth[x]*Csch[x]) + ((8 *I)*Coth[x]*Csch[x])/(1 - I*Sinh[x]))/3 + ((I/3)*Coth[x]*Csch[x])/(1 - I*S inh[x])^2)
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (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 = 0.96 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {-98 \,{\mathrm e}^{4 x}+63 i {\mathrm e}^{5 x}+97 \,{\mathrm e}^{2 x}-126 i {\mathrm e}^{3 x}+21 \,{\mathrm e}^{6 x}-32+75 i {\mathrm e}^{x}}{3 \left ({\mathrm e}^{2 x}-1\right )^{2} \left ({\mathrm e}^{x}+i\right )^{3}}+\frac {7 \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {7 \ln \left ({\mathrm e}^{x}+1\right )}{2}\) | \(72\) |
| default | \(i \tanh \left (\frac {x}{2}\right )-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}+\frac {8 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}+\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {i}{\tanh \left (\frac {x}{2}\right )}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}\) | \(76\) |
| parallelrisch | \(\frac {\left (252 i \tanh \left (\frac {x}{2}\right )^{2}+84 \tanh \left (\frac {x}{2}\right )^{3}-84 i-252 \tanh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )+15 i \tanh \left (\frac {x}{2}\right )^{4}-3 \tanh \left (\frac {x}{2}\right )^{5}-3 i \coth \left (\frac {x}{2}\right )^{2}-112 \tanh \left (\frac {x}{2}\right )^{3}-190 i+15 \coth \left (\frac {x}{2}\right )-234 \tanh \left (\frac {x}{2}\right )}{-72 \tanh \left (\frac {x}{2}\right )+72 i \tanh \left (\frac {x}{2}\right )^{2}+24 \tanh \left (\frac {x}{2}\right )^{3}-24 i}\) | \(111\) |
Input:
int(csch(x)^3/(I+sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
1/3*(-98*exp(x)^4+63*I*exp(x)^5+97*exp(x)^2-126*I*exp(x)^3+21*exp(x)^6-32+ 75*I*exp(x))/(exp(x)^2-1)^2/(exp(x)+I)^3+7/2*ln(exp(x)-1)-7/2*ln(exp(x)+1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (40) = 80\).
Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.00 \[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {21 \, {\left (e^{\left (7 \, x\right )} + 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} - 7 i \, e^{\left (4 \, x\right )} + 7 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 21 \, {\left (e^{\left (7 \, x\right )} + 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} - 7 i \, e^{\left (4 \, x\right )} + 7 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 42 \, e^{\left (6 \, x\right )} - 126 i \, e^{\left (5 \, x\right )} + 196 \, e^{\left (4 \, x\right )} + 252 i \, e^{\left (3 \, x\right )} - 194 \, e^{\left (2 \, x\right )} - 150 i \, e^{x} + 64}{6 \, {\left (e^{\left (7 \, x\right )} + 3 i \, e^{\left (6 \, x\right )} - 5 \, e^{\left (5 \, x\right )} - 7 i \, e^{\left (4 \, x\right )} + 7 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \] Input:
integrate(csch(x)^3/(I+sinh(x))^2,x, algorithm="fricas")
Output:
-1/6*(21*(e^(7*x) + 3*I*e^(6*x) - 5*e^(5*x) - 7*I*e^(4*x) + 7*e^(3*x) + 5* I*e^(2*x) - 3*e^x - I)*log(e^x + 1) - 21*(e^(7*x) + 3*I*e^(6*x) - 5*e^(5*x ) - 7*I*e^(4*x) + 7*e^(3*x) + 5*I*e^(2*x) - 3*e^x - I)*log(e^x - 1) - 42*e ^(6*x) - 126*I*e^(5*x) + 196*e^(4*x) + 252*I*e^(3*x) - 194*e^(2*x) - 150*I *e^x + 64)/(e^(7*x) + 3*I*e^(6*x) - 5*e^(5*x) - 7*I*e^(4*x) + 7*e^(3*x) + 5*I*e^(2*x) - 3*e^x - I)
\[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \] Input:
integrate(csch(x)**3/(I+sinh(x))**2,x)
Output:
Integral(csch(x)**3/(sinh(x) + I)**2, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (40) = 80\).
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.81 \[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=-\frac {-75 i \, e^{\left (-x\right )} + 97 \, e^{\left (-2 \, x\right )} + 126 i \, e^{\left (-3 \, x\right )} - 98 \, e^{\left (-4 \, x\right )} - 63 i \, e^{\left (-5 \, x\right )} + 21 \, e^{\left (-6 \, x\right )} - 32}{3 \, {\left (3 \, e^{\left (-x\right )} + 5 i \, e^{\left (-2 \, x\right )} - 7 \, e^{\left (-3 \, x\right )} - 7 i \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )} + 3 i \, e^{\left (-6 \, x\right )} - e^{\left (-7 \, x\right )} - i\right )}} - \frac {7}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {7}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate(csch(x)^3/(I+sinh(x))^2,x, algorithm="maxima")
Output:
-1/3*(-75*I*e^(-x) + 97*e^(-2*x) + 126*I*e^(-3*x) - 98*e^(-4*x) - 63*I*e^( -5*x) + 21*e^(-6*x) - 32)/(3*e^(-x) + 5*I*e^(-2*x) - 7*e^(-3*x) - 7*I*e^(- 4*x) + 5*e^(-5*x) + 3*I*e^(-6*x) - e^(-7*x) - I) - 7/2*log(e^(-x) + 1) + 7 /2*log(e^(-x) - 1)
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=\frac {e^{\left (3 \, x\right )} + 4 i \, e^{\left (2 \, x\right )} + e^{x} - 4 i}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + \frac {2 \, {\left (9 \, e^{\left (2 \, x\right )} + 21 i \, e^{x} - 10\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} - \frac {7}{2} \, \log \left (e^{x} + 1\right ) + \frac {7}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(csch(x)^3/(I+sinh(x))^2,x, algorithm="giac")
Output:
(e^(3*x) + 4*I*e^(2*x) + e^x - 4*I)/(e^(2*x) - 1)^2 + 2/3*(9*e^(2*x) + 21* I*e^x - 10)/(e^x + I)^3 - 7/2*log(e^x + 1) + 7/2*log(abs(e^x - 1))
Time = 2.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.36 \[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {7\,\ln \left ({\mathrm {e}}^x+1\right )}{2}-\frac {7\,\ln \left (\frac {1}{{\mathrm {e}}^x-1}\right )}{2}+\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {6}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {2{}\mathrm {i}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^2}+\frac {4}{3\,{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^3}+\frac {4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \] Input:
int(1/(sinh(x)^3*(sinh(x) + 1i)^2),x)
Output:
exp(x)/(exp(2*x) - 1) - (7*log(exp(x) + 1))/2 - (7*log(1/(exp(x) - 1)))/2 + (2*exp(x))/(exp(2*x) - 1)^2 + 6/(exp(x) + 1i) + 2i/(exp(x) + 1i)^2 + 4/( 3*(exp(x) + 1i)^3) + 4i/(exp(2*x) - 1)
\[ \int \frac {\text {csch}^3(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\mathrm {csch}\left (x \right )^{3}}{\sinh \left (x \right )^{2}+2 \sinh \left (x \right ) i -1}d x \] Input:
int(csch(x)^3/(I+sinh(x))^2,x)
Output:
int(csch(x)**3/(sinh(x)**2 + 2*sinh(x)*i - 1),x)