Integrand size = 13, antiderivative size = 64 \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=-5 i \text {arctanh}(\cosh (x))-12 \coth (x)+4 \coth ^3(x)+5 i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))} \] Output:
-5*I*arctanh(cosh(x))-12*coth(x)+4*coth(x)^3+5*I*coth(x)*csch(x)+1/3*coth( x)*csch(x)^2/(I+sinh(x))^2-10/3*I*coth(x)*csch(x)^2/(I+sinh(x))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(64)=128\).
Time = 1.98 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.23 \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{24} \left (-44 \coth \left (\frac {x}{2}\right )+6 i \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{2} \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+2 \left (-60 i \log \left (\cosh \left (\frac {x}{2}\right )\right )+60 i \log \left (\sinh \left (\frac {x}{2}\right )\right )+3 i \text {sech}^2\left (\frac {x}{2}\right )-4 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {4}{i+\sinh (x)}+\frac {8 \sinh \left (\frac {x}{2}\right ) (14 i+13 \sinh (x))}{\left (i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )^3}-22 \tanh \left (\frac {x}{2}\right )\right )\right ) \] Input:
Integrate[Csch[x]^4/(I + Sinh[x])^2,x]
Output:
(-44*Coth[x/2] + (6*I)*Csch[x/2]^2 + (Csch[x/2]^4*Sinh[x])/2 + 2*((-60*I)* Log[Cosh[x/2]] + (60*I)*Log[Sinh[x/2]] + (3*I)*Sech[x/2]^2 - 4*Csch[x]^3*S inh[x/2]^4 - 4/(I + Sinh[x]) + (8*Sinh[x/2]*(14*I + 13*Sinh[x]))/(I*Cosh[x /2] + Sinh[x/2])^3 - 22*Tanh[x/2]))/24
Time = 0.70 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.38, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.385, Rules used = {3042, 3245, 27, 3042, 3457, 27, 3042, 3227, 26, 3042, 26, 4254, 2009, 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.
| \(\displaystyle \int \frac {\text {csch}^4(x)}{(\sinh (x)+i)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(i-i \sin (i x))^2 \sin (i x)^4}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {1}{3} \int \frac {2 \text {csch}^4(x) (3 i-2 \sinh (x))}{\sinh (x)+i}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \int \frac {\text {csch}^4(x) (3 i-2 \sinh (x))}{\sinh (x)+i}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \int \frac {2 i \sin (i x)+3 i}{(i-i \sin (i x)) \sin (i x)^4}dx\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}-\int -3 \text {csch}^4(x) (5 i \sinh (x)+6)dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \int \text {csch}^4(x) (5 i \sinh (x)+6)dx+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \int \frac {5 \sin (i x)+6}{\sin (i x)^4}dx+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (6 \int \text {csch}^4(x)dx+5 \int i \text {csch}^3(x)dx\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (6 \int \text {csch}^4(x)dx+5 i \int \text {csch}^3(x)dx\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 i \int -i \csc (i x)^3dx+6 \int \csc (i x)^4dx\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \int \csc (i x)^3dx+6 \int \csc (i x)^4dx\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \int \csc (i x)^3dx+6 i \int \left (1-\coth ^2(x)\right )d(-i \coth (x))\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \int \csc (i x)^3dx+6 i \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \left (\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+6 i \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \left (-\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+6 i \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \left (-\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+6 i \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \left (\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+6 i \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2}-\frac {2}{3} \left (3 \left (5 \left (\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+6 i \left (\frac {1}{3} i \coth ^3(x)-i \coth (x)\right )\right )+\frac {5 i \coth (x) \text {csch}^2(x)}{\sinh (x)+i}\right )\) |
Input:
Int[Csch[x]^4/(I + Sinh[x])^2,x]
Output:
(Coth[x]*Csch[x]^2)/(3*(I + Sinh[x])^2) - (2*(3*((6*I)*((-I)*Coth[x] + (I/ 3)*Coth[x]^3) + 5*((I/2)*ArcTanh[Cosh[x]] - (I/2)*Coth[x]*Csch[x])) + ((5* I)*Coth[x]*Csch[x]^2)/(I + Sinh[x])))/3
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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 1.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.38
| method | result | size |
| risch | \(\frac {2 i \left (45 i {\mathrm e}^{7 x}+15 \,{\mathrm e}^{8 x}-135 i {\mathrm e}^{5 x}-85 \,{\mathrm e}^{6 x}+155 i {\mathrm e}^{3 x}+153 \,{\mathrm e}^{4 x}-57 i {\mathrm e}^{x}-99 \,{\mathrm e}^{2 x}+24\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3} \left ({\mathrm e}^{x}+i\right )^{3}}+5 i \ln \left ({\mathrm e}^{x}-1\right )-5 i \ln \left ({\mathrm e}^{x}+1\right )\) | \(88\) |
| default | \(-\frac {15 \tanh \left (\frac {x}{2}\right )}{8}+\frac {\tanh \left (\frac {x}{2}\right )^{3}}{24}-\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{4}+\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {10}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{4 \tanh \left (\frac {x}{2}\right )^{2}}+5 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {15}{8 \tanh \left (\frac {x}{2}\right )}\) | \(92\) |
| parallelrisch | \(\frac {\left (-360 i \tanh \left (\frac {x}{2}\right )^{2}-120 \tanh \left (\frac {x}{2}\right )^{3}+120 i+360 \tanh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )+i \tanh \left (\frac {x}{2}\right )^{6}-30 i \tanh \left (\frac {x}{2}\right )^{4}+3 \tanh \left (\frac {x}{2}\right )^{5}+3 i \coth \left (\frac {x}{2}\right )^{2}+\coth \left (\frac {x}{2}\right )^{3}+170 \tanh \left (\frac {x}{2}\right )^{3}+290 i-30 \coth \left (\frac {x}{2}\right )+360 \tanh \left (\frac {x}{2}\right )}{-72 \tanh \left (\frac {x}{2}\right )^{2}+24 i \tanh \left (\frac {x}{2}\right )^{3}+24-72 i \tanh \left (\frac {x}{2}\right )}\) | \(126\) |
Input:
int(csch(x)^4/(I+sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
2/3*I*(45*I*exp(x)^7+15*exp(x)^8-135*I*exp(x)^5-85*exp(x)^6+155*I*exp(x)^3 +153*exp(x)^4-57*I*exp(x)-99*exp(x)^2+24)/(exp(x)^2-1)^3/(exp(x)+I)^3+5*I* ln(exp(x)-1)-5*I*ln(exp(x)+1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (50) = 100\).
Time = 0.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.53 \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {15 \, {\left (i \, e^{\left (9 \, x\right )} - 3 \, e^{\left (8 \, x\right )} - 6 i \, e^{\left (7 \, x\right )} + 10 \, e^{\left (6 \, x\right )} + 12 i \, e^{\left (5 \, x\right )} - 12 \, e^{\left (4 \, x\right )} - 10 i \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + 15 \, {\left (-i \, e^{\left (9 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 6 i \, e^{\left (7 \, x\right )} - 10 \, e^{\left (6 \, x\right )} - 12 i \, e^{\left (5 \, x\right )} + 12 \, e^{\left (4 \, x\right )} + 10 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 3 i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 30 i \, e^{\left (8 \, x\right )} + 90 \, e^{\left (7 \, x\right )} + 170 i \, e^{\left (6 \, x\right )} - 270 \, e^{\left (5 \, x\right )} - 306 i \, e^{\left (4 \, x\right )} + 310 \, e^{\left (3 \, x\right )} + 198 i \, e^{\left (2 \, x\right )} - 114 \, e^{x} - 48 i}{3 \, {\left (e^{\left (9 \, x\right )} + 3 i \, e^{\left (8 \, x\right )} - 6 \, e^{\left (7 \, x\right )} - 10 i \, e^{\left (6 \, x\right )} + 12 \, e^{\left (5 \, x\right )} + 12 i \, e^{\left (4 \, x\right )} - 10 \, e^{\left (3 \, x\right )} - 6 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + i\right )}} \] Input:
integrate(csch(x)^4/(I+sinh(x))^2,x, algorithm="fricas")
Output:
-1/3*(15*(I*e^(9*x) - 3*e^(8*x) - 6*I*e^(7*x) + 10*e^(6*x) + 12*I*e^(5*x) - 12*e^(4*x) - 10*I*e^(3*x) + 6*e^(2*x) + 3*I*e^x - 1)*log(e^x + 1) + 15*( -I*e^(9*x) + 3*e^(8*x) + 6*I*e^(7*x) - 10*e^(6*x) - 12*I*e^(5*x) + 12*e^(4 *x) + 10*I*e^(3*x) - 6*e^(2*x) - 3*I*e^x + 1)*log(e^x - 1) - 30*I*e^(8*x) + 90*e^(7*x) + 170*I*e^(6*x) - 270*e^(5*x) - 306*I*e^(4*x) + 310*e^(3*x) + 198*I*e^(2*x) - 114*e^x - 48*I)/(e^(9*x) + 3*I*e^(8*x) - 6*e^(7*x) - 10*I *e^(6*x) + 12*e^(5*x) + 12*I*e^(4*x) - 10*e^(3*x) - 6*I*e^(2*x) + 3*e^x + I)
Timed out. \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=\text {Timed out} \] Input:
integrate(csch(x)**4/(I+sinh(x))**2,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (50) = 100\).
Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.98 \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (57 \, e^{\left (-x\right )} + 99 i \, e^{\left (-2 \, x\right )} - 155 \, e^{\left (-3 \, x\right )} - 153 i \, e^{\left (-4 \, x\right )} + 135 \, e^{\left (-5 \, x\right )} + 85 i \, e^{\left (-6 \, x\right )} - 45 \, e^{\left (-7 \, x\right )} - 15 i \, e^{\left (-8 \, x\right )} - 24 i\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} + 6 i \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-3 \, x\right )} - 12 i \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )} + 10 i \, e^{\left (-6 \, x\right )} - 6 \, e^{\left (-7 \, x\right )} - 3 i \, e^{\left (-8 \, x\right )} + e^{\left (-9 \, x\right )} - i\right )}} - 5 i \, \log \left (e^{\left (-x\right )} + 1\right ) + 5 i \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate(csch(x)^4/(I+sinh(x))^2,x, algorithm="maxima")
Output:
-2/3*(57*e^(-x) + 99*I*e^(-2*x) - 155*e^(-3*x) - 153*I*e^(-4*x) + 135*e^(- 5*x) + 85*I*e^(-6*x) - 45*e^(-7*x) - 15*I*e^(-8*x) - 24*I)/(3*e^(-x) + 6*I *e^(-2*x) - 10*e^(-3*x) - 12*I*e^(-4*x) + 12*e^(-5*x) + 10*I*e^(-6*x) - 6* e^(-7*x) - 3*I*e^(-8*x) + e^(-9*x) - I) - 5*I*log(e^(-x) + 1) + 5*I*log(e^ (-x) - 1)
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.31 \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=-\frac {2 \, {\left (-15 i \, e^{\left (8 \, x\right )} + 45 \, e^{\left (7 \, x\right )} + 85 i \, e^{\left (6 \, x\right )} - 135 \, e^{\left (5 \, x\right )} - 153 i \, e^{\left (4 \, x\right )} + 155 \, e^{\left (3 \, x\right )} + 99 i \, e^{\left (2 \, x\right )} - 57 \, e^{x} - 24 i\right )}}{3 \, {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )}^{3}} - 5 i \, \log \left (e^{x} + 1\right ) + 5 i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(csch(x)^4/(I+sinh(x))^2,x, algorithm="giac")
Output:
-2/3*(-15*I*e^(8*x) + 45*e^(7*x) + 85*I*e^(6*x) - 135*e^(5*x) - 153*I*e^(4 *x) + 155*e^(3*x) + 99*I*e^(2*x) - 57*e^x - 24*I)/(e^(3*x) + I*e^(2*x) - e ^x - I)^3 - 5*I*log(e^x + 1) + 5*I*log(abs(e^x - 1))
Time = 2.39 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.95 \[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=-\ln \left (-{\mathrm {e}}^x\,10{}\mathrm {i}-10{}\mathrm {i}\right )\,5{}\mathrm {i}+\ln \left (-{\mathrm {e}}^x\,10{}\mathrm {i}+10{}\mathrm {i}\right )\,5{}\mathrm {i}-\frac {\frac {16\,{\mathrm {e}}^x}{3}-\frac {{\mathrm {e}}^{2\,x}\,32{}\mathrm {i}}{3}+\frac {16}{3}{}\mathrm {i}}{12\,{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,12{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,6{}\mathrm {i}-{\mathrm {e}}^{6\,x}\,10{}\mathrm {i}-6\,{\mathrm {e}}^{7\,x}+{\mathrm {e}}^{8\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{9\,x}+3\,{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {\frac {20\,{\mathrm {e}}^{2\,x}}{3}-\frac {44}{3}+\frac {{\mathrm {e}}^x\,16{}\mathrm {i}}{3}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^{5\,x}\,2{}\mathrm {i}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {10\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+\frac {20}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}+{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x-\mathrm {i}} \] Input:
int(1/(sinh(x)^4*(sinh(x) + 1i)^2),x)
Output:
log(10i - exp(x)*10i)*5i - log(- exp(x)*10i - 10i)*5i - ((16*exp(x))/3 - ( exp(2*x)*32i)/3 + 16i/3)/(exp(4*x)*12i - 10*exp(3*x) - exp(2*x)*6i + 12*ex p(5*x) - exp(6*x)*10i - 6*exp(7*x) + exp(8*x)*3i + exp(9*x) + 3*exp(x) + 1 i) + ((20*exp(2*x))/3 + (exp(x)*16i)/3 - 44/3)/(3*exp(2*x) - exp(3*x)*4i - 3*exp(4*x) + exp(5*x)*2i + exp(6*x) + exp(x)*2i - 1) - (10*exp(x) - exp(2 *x)*10i + 20i/3)/(exp(2*x)*1i + exp(3*x) - exp(x) - 1i)
\[ \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx=\int \frac {\mathrm {csch}\left (x \right )^{4}}{\sinh \left (x \right )^{2}+2 \sinh \left (x \right ) i -1}d x \] Input:
int(csch(x)^4/(I+sinh(x))^2,x)
Output:
int(csch(x)**4/(sinh(x)**2 + 2*sinh(x)*i - 1),x)